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(2e) and Its Galois Extension
Finite Fields and Their Applications 8, 570–588 (2002) doi:10.1006/ffta.2002.0365
Compressing Mappings on Primitive Sequences over Z=ð2e Þ and Its Galois Extension1 Qi Wenfeng2 and Zhu Xuanyong Department of Applied Mathematics, Zhengzhou Information Engineering University, P.O. Box 1001-745, Zhengzhou, 450002, People’s Republic of China E-mail: [email protected],[email protected] Communicated by Rudolf Lidl Received March 11, 2001; revised February 10, 2002; published online June 25, 2002
Let f ðxÞ be a strongly primitive polynomial of degree n over Z=ð2e Þ, Zðx0 ; x1 ; . . . ; xe2 Þ a Boolean function of e 1 variables and jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ Gðf ðxÞ; Z=ð2e ÞÞ denotes the set of all sequences over Z=ð2e Þ generated by f ðxÞ, F21 the set of all sequences over the binary field F2 , then the compressing mapping ( F:
Gðf ðxÞ; Z=ð2e ÞÞ ! F21 ; a ¼ a0 þ a1 2 þ þ ae1 2e1 /jða0 ; a1 ; . . . ; ae1 Þ mod 2 % % % % % % %
is injective, that is, for a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ, a ¼ b if and only if FðaÞ ¼ FðbÞ, i.e., % part of the paper, % we general% jða0 ; . . . ; ae1 Þ ¼ jðb0 ; . .%. ;%be1 Þ mod 2. In the %second % % Galois rings. # 2002 Elsevier Science (USA) ize% the above result %over the Key Words: primitive polynomial; Galois ring; linear sequence; compressing mapping.
1.
INTRODUCTION
Let R be a ring, f ðxÞ ¼ xn þ cn1 xn1 þ þ c0 a monic polynomial over R, the sequence a ¼ ða0 ; a1 ; a2 ; . . .Þ over R satisfying the recursion % aiþn ¼ ðc0 ai þ c1 aiþ1 þ þ cn1 aiþn1 Þ; i ¼ 0; 1; 2; . . . 1 The work is supported by HAIPURT and the Special Fund of National Excellently Doctoral Paper. 2 To whom correspondence should be addressed.
570 1071-5797/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
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is called a linear recurring sequence of degree n over R, generated by f ðxÞ. We will use the notation Gðf ðxÞ; RÞ for the set of all sequences over R generated by f ðxÞ. Let a ¼ ða0 ; a1 ; a2 ; . . .Þ and b ¼ ðb0 ; b1 ; b2 ; . . .Þ be sequences over R and % % c 2 R; define a þ b ¼ ða0 þ b0 ; a1 þ b1 ; a2 þ b2 ; . . .Þ; % % ca ¼ ðca0 ; ca1 ; ca2 ; . . .Þ; % ab ¼ ða0 b0 ; a1 b1 ; a2 b2 ; . . .Þ %% and the shift operator x of sequence as xa ¼ ða1 ; a2 ; a3 ; . . .Þ and xk a ¼ ðak ; akþ1 ; akþ2 ; . . .Þ for k ¼ 0; 1; 2; . . . : Then for% any polynomial f ðxÞ over% R, a 2 Gðf ðxÞ; RÞ if and only if f ðxÞa ¼ 0. % Let f ðxÞ be a monic polynomial % of degree n over Z=ð2e Þ with f ð0Þc0 mod 2, then there exists a positive integer P such that f ðxÞ divides xP 1 over Z=ð2e Þ. The least such P is called the period of f ðxÞ over Z=ð2e Þ and denoted by perðf ðxÞÞ. The period of f ðxÞ is upper bounded by 2e1 ð2n 1Þ, where n ¼ deg f ðxÞ. A monic polynomial f ðxÞ of degree n over Z=ð2e Þ is called a primitive polynomial if perðf ðxÞÞ ¼ 2e1 ð2n 1Þ. Let f ðxÞ be a primitive polynomial of degree n over Z=ð2e Þ, T ¼ 2n 1, then f ðxÞ mod 2 is a primitive polynomial over the binary field F2 and for i ¼ 1; 2; . . . ; e 1, we have x2
i1
T
1 2i hi ðxÞ mod f ðxÞ;
ð1Þ
where hi ðxÞ is a polynomial over Z=ð2e Þ of degree less than n such that hi ðxÞc0 mod 2. Furthermore, h2 ðxÞ he1 ðxÞ mod 2; h2 ðxÞ h1 ðxÞ þ h1 ðxÞ2 mod ð2; f ðxÞÞ:
ð2Þ
If e 3 and h2 ðxÞc1 mod 2 or e ¼ 2 and h1 ðxÞc1 mod 2; then f ðxÞ is called a strongly primitive polynomial over Z=ð2e Þ (see [1, 3]). Any element a in Z=ð2e Þ has a unique binary decomposition as a ¼ a0 þ a1 2 þ þ ae1 2e1 , ai 2 f0; 1g. Similarly, a sequence a over Z=ð2e Þ has a unique binary decomposition as a ¼ a0 þ a1 2 þ þ ae1% 2e1 , where % over % %f0; 1g. The % sequence a is ai ¼ ðai0 ; ai1 ; ai2 ; . . .Þ is a binary sequence i %called ith level component of a, and a % e1 the highest level component of a. % % properties of the level component of % There are many papers to discuss the a, please refer to [1–5, 7, 8]. %
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In the first part of the paper, we prove the following result. Let f ðxÞ be a strongly primitive polynomial of degree n over Z=ð2e Þ, Zðx0 ; x1 ; . . . ; xe2 Þ a Boolean function of e 1 variables and jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ: F21 denotes the set of all sequences over the binary field F2 ; then the compressing map ( Gðf ðxÞ; Z=ð2e ÞÞ ! F21 ; F: a ¼ a0 þ a1 2 þ þ ae1 2e1 /jða0 ; a1 ; . . . ; ae1 Þ mod 2 % % % % % % % is injective, that is, for a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ, a ¼ b if and only if % Þ% mod 2. % % jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; be1 %In the second % % % part of the paper, we generalize the above result to the one over Galois rings.
2. INJECTIVENESS OF COMPRESSION MAPPINGS OVER Z=ð2e Þ Huang [3] and Huang and Dai [4] proposed the following injectiveness theorem. Theorem 1. mapping
Let f ðxÞ be a primitive polynomial over Z=ð2e Þ, then the ( F:
Gðf ðxÞ; Z=ð2e ÞÞ ! F21 ; a ¼ a0 þ a1 2 þ þ ae1 2e1 /ae1 % % % % %
is injective, that is, for a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ; a ¼ b if and only if ae1 ¼ be1 . % % % % % % Remark 1. Theorem 1 implies that ae1 contains all information of the % original sequence a. % Lemma 1 (Dai [1]). Let f ðxÞ be a primitive polynomial of degree n over Z=ð2e Þ, T ¼ 2n 1, a ¼ a0 þ a1 2 þ þ ae1 2e1 2 Gðf ðxÞ; Z=ð2e ÞÞ and % % perða Þ% ¼ 2e1 T ¼ perðaÞ ¼ perðf ðxÞÞ: a0 =0, then perðai Þ ¼ 2i T% , especially e1 % % % % Lemma 2 (Dai [1]). Let f ðxÞ be a primitive polynomial of degree n over Z=ð2e Þ, T ¼ 2n 1; a ¼ a0 þ a1 2 þ þ ae1 2e1 2 Gðf ðxÞ; Z=ð2e ÞÞ, then i1 % %2 for% i ¼ 1; 2; . . .%; e 1. That is, by (2), ðx2 T 1Þai hi ðxÞa0 mod % % 8 if i ¼ 1; < h1 ðxÞa0 mod 2 i1 % ðx2 T 1Þai
: h2 ðxÞa0 mod 2 if i 2: % %
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
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Lemma 3. Let S be a positive integer, u a sequence over the ring R with its period dividing S, then for any sequence v %over R, ðxS 1Þðv uÞ ¼ uðxS 1Þv: % %% % % The proof is easy. Lemma 4. Let f ðxÞ be a primitive polynomial of degree n over the finite field F2r ; and a ¼ ða0 ; a1 ; . . .Þ, b ¼ ðb0 ; b1 ; . . .Þ, c ¼ ðc0 ; c1 ; . . .Þ 2 Gðf ðxÞ; F2r Þ. % linear independent % If a, b and c are over F2r , %then the number of zeros in % % % fa0 b0 ; a1 b1 ; . . . ; aT 1 bT 1 g is smaller than the one in fa0 b0 c0 ; a1 b1 c1 ; . . . ; aT 1 bT 1 cT 1 g; where T ¼ 2nr 1. Thus a b=a b c: %% %%% The proof is easy. Lemma 5. Let f ðxÞ be a primitive polynomial of degree n over Z=ð2e Þ, Zðx0 ; x1 ; . . . ; xe2 Þ a Boolean function of e 1 variables and jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ: If jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; be1 Þ mod 2 for a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ; then % % % % % a0 ¼ b%0 . % % Proof. Set T ¼ 2n 1. The periods of Zða0 ; . . . ; ae2 Þ and Zðb0 ; . . . ; be2 Þ e2 % a ; . . .%; a Þ jðb% ; . . . ; b% Þ divide 2e2 T , so, by x2 T 1 acting on jð 0 e1 0 e1 % % % % mod 2, we get ðx2
e2
T
1Þae1 ðx2 %
e2
T
1Þbe1 mod 2: %
Since perðai Þ ¼ 2i T for 0 i e 2, by x2e2T 1 2e1 he1 ðxÞ mod f ðxÞ acting on %a and b, respectively, we have % % e2 ðx2 T 1Þae1 2e1 ¼ 2e1 he1 ðxÞa0 ; % % ðx2
e2
that is,
T
1Þbe1 2e1 ¼ 2e1 he1 ðxÞb0 ; % %
ðx2
e2
T
1Þae1 he1 ðxÞa0 mod 2; % %
ðx2
e2
T
1Þbe1 he1 ðxÞb0 mod 2: % %
So he1 ðxÞa0 he1 ðxÞb0 mod 2: Since he1 ðxÞc0 modð2; f ðxÞÞ; we have % a0 ¼ b0 : &% % % Theorem 2. Let f ðxÞ be a strongly primitive polynomial of degree n over Z=ð2e Þ, Zðx0 ; x1 ; . . . ; xe2 Þ a Boolean function of e 1 variables and jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ;
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then the compressing map ( F:
Gðf ðxÞ; Z=ð2e ÞÞ ! F21 ; a ¼ a0 þ a1 2 þ þ ae1 2e1 /jða0 ; a1 ; . . . ; ae1 Þ mod 2 % % % % % % %
is injective, that is, for a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ, a ¼ b if and only if jða0 ; . . . ; % % 2: % ae1 Þ jðb0 ; . . . ; be1 Þ mod % % % Proof. Let a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ with jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; be1 Þ % 2, we% set mod 2 we shall% %prove a ¼ b. Since h2 ðxÞ % %he1 ðxÞ mod % % hðxÞ ¼ h2 ðxÞ mod 2 which is a polynomial over the field F2 : Since a0 ¼ b0 by Lemma 5, it suffices to consider the case a0 ¼ b0 =0. Set % % Ze2 ðx0 ; x%1 ; . . . %; xe2 Þ ¼ Zðx0 ; x1 ; . . . ; xe2 Þ, then Ze2 ðx0 ; x1 ; . . . ; xe2 Þ ¼ xe2 Ze3 ðx0 ; x1 ; . . . ; xe3 Þ þ ce3 ðx0 ; x1 ; . . . ; xe3 Þ: In general, we have Zi ðx0 ; x1 ; . . . ; xi Þ ¼ xi Zi1 ðx0 ; x1 ; . . . ; xi1 Þ þ ci1 ðx0 ; x1 ; . . . ; xi1 Þ;
ð3Þ
where Zi1 ðx0 ; x1 ; . . . ; xi1 Þ and ci1 ðx0 ; x1 ; . . . ; xi1 Þ are Boolean functions with i variables, i ¼ 1; 2; . . . ; e 1: Firstly, we consider the case e 4: Since ae1 þ ae2 Ze3 ða0 ; . . . ; ae3 Þ þ ce3 ða0 ; . . . ; ae3 Þ % % % % % %
be1 þ be2 Ze3 ðb0 ; . . . ; be3 Þ þ ce3 ðb0 ; . . . ; be3 Þ mod 2 % % % % % % and the periods of ce3 ða0 ; . . . ; ae3 Þ and ce3 ðb0 ; . . . ; be3 Þ divide 2e3 T , it % % % % follows ðx2
e3
ðx2 and
ðx2
e3
T
e3
T
1Þce3 ða0 ; . . . ; ae3 Þ 0 mod 2; % % 1Þce3 ðb0 ; . . . ; be3 Þ 0 mod 2 % %
T
1Þðae1 þ ae2 Ze3 ða0 ; . . . ; ae3 ÞÞ % % % % 2 T
ðx 1Þðbe1 þ be2 Ze3 ðb0 ; . . . ; be3 ÞÞ mod 2: % % % % e3
ð4Þ
The periods of Ze3 ða0 ; a1 ; . . . ; ae3 Þ and Ze3 ðb0 ; b1 ; . . . ; be3 Þ divide 2e3 T e3 % hðxÞ % a %hðxÞb ðx2e3 T% %1Þb %mod 2, so (4) and ðx2 T 1Þae2
0 0 e2 % % % %
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
575
implies ðx2
e3
T
1Þae1 þ Ze3 ða0 ; . . . ; ae3 ÞhðxÞa0 % % % % 2e3 T
ðx 1Þbe1 þ Ze3 ðb0 ; . . . ; be3 ÞhðxÞa0 mod 2; % % % %
that is, ðx2
e3
T
1Þðae1 þ be1 Þ % %
½Ze3 ða0 ; . . . ; ae3 Þ þ Ze3 ðb0 ; . . . ; be3 ÞhðxÞa0 mod 2: % % % % %
ð5Þ
e3
On the other hand, by x2 T 1 2e2 he2 ðxÞ mod f ðxÞ acting on a ¼ % a0 þ a1 2 þ þ ae1 2e1 ; we get % % % e3 ðx2 T 1Þðae2 þ ae1 2Þ2e2 2e2 he2 ðxÞða0 þ a1 2Þ; % % % % that is, ðx2
e3
T
1Þðae2 þ ae1 2Þ he2 ðxÞða0 þ a1 2Þ mod 22 : % % % %
ð6Þ
Let he2 ðxÞða0 þ a1 2Þ u þ v2 mod 22 , where u and v are the 0th and 1th % % of h % ðxÞð%a þ a 2Þ, respectively, % % level components then u hðxÞa0 mod 2. e2 0 1 % % % % By (6), e3
x2
T
ae2 þ ðx2 %
e3
T
1Þae1 2 ae2 þ u þ v2 mod 22 % % % %
and so e3
ðx2
T
1Þae1 v þ ae2 u mod 2: % % % %
ð7Þ
T
1Þbe1 w þ be2 u mod 2; % % % %
ð8Þ
Similarly, e3
ðx2
where he2 ðxÞðb0 þ b1 2Þ u þ w2 mod 22 and u he2 ðxÞb0 hðxÞa0 mod 2. % a% þ b Þ mod 2: By % (7) and %(8), % Since a0 ¼ b%0 , v þ% w hðxÞð 1 1 % % % % % % e3 ðx2 T 1Þðae1 þ be1 Þ ðae2 þ be2 ÞhðxÞa0 þ hðxÞða1 þ b1 Þ mod 2: ð9Þ % % % % % % % Comparing with (5), it follows that ðae2 þ be2 ÞhðxÞa0 % % %
hðxÞða1 þ b1 Þ½Ze3 ða0 ; . . . ; ae3 Þ þ Ze3 ðb0 ; . . . ; be3 ÞhðxÞa0 mod 2: % % % % % % %
ð10Þ
576
WENFENG AND XUANYONG e4
If e 5, then x2 T 1 2e3 he3 ðxÞ mod f ðxÞ acts on a ¼ a0 þ a1 2 þ % % % þ ae1 2e1 and b ¼ b0 þ b1 2 þ þ be1 2e1 . It follows that % % % % % e4 ð11Þ ðx2 T 1Þðae3 þ ae2 2Þ he3 ðxÞða0 þ a1 2Þ mod 22 ; % % % % e4 ð12Þ ðx2 T 1Þðbe3 þ be2 2Þ he3 ðxÞðb0 þ b1 2Þ mod 22 : % % % % Similar to (9), we get e4
ðx2
T
1Þðae2 þ be2 Þ ðae3 þ be3 ÞhðxÞa0 þ hðxÞða1 þ b1 Þ mod 2: % % % % % % %
Multiplying (13) by hðxÞa0 , since perðhðxÞa0 Þ divides T ; we obtain % % e4 ðx2 T 1Þ½ðae2 þ be2 ÞhðxÞa0 % % %
ðae3 þ be3 ÞhðxÞa0 þ hðxÞða1 þ b1 ÞhðxÞa0 mod 2: % % % % % %
ð13Þ
ð14Þ
By (10) and (3), we have ðae2 þ be2 ÞhðxÞa0 hðxÞða1 þ b1 Þ½ae3 Ze4 ða0 ; . . . ; ae4 Þ % % % % % % % % þ be3 Ze4 ðb0 ; . . . ; be4 Þ þ ce4 ða0 ; . . . ; ae4 Þ % % % % % þ ce4 ðb0 ; . . . ; be4 ÞhðxÞa0 mod 2: % % % It follows that e4
ðx2
T
1Þ½ðae2 þ be2 ÞhðxÞa0 % % % 2e4 T
ðx 1Þ½ae3 Ze4 ða0 ; . . . ; ae4 Þ þ be3 Ze4 ðb0 ; . . . ; be4 ÞhðxÞa0 % % % % % % %
½Ze4 ða0 ; . . . ; ae4 Þ þ Ze4 ðb0 ; . . . ; be4 ÞhðxÞa0 mod 2: % % % % %
Comparing with (14), we get ðae3 þ be3 ÞhðxÞa0 hðxÞða1 þ b1 ÞhðxÞa0 þ ½Ze4 ða0 ; . . . ; ae4 Þ % % % % % % % % þ Ze4 ðb0 ; . . . ; be4 ÞhðxÞa0 mod 2: % % %
ð15Þ
Now, we show hðxÞða1 þ b1 ÞhðxÞa0 hðxÞða1 þ b1 Þ mod 2. If a1 ¼ b1 , then % % % % % % % hðxÞða1 þ b1 ÞhðxÞa0 hðxÞða1 þ b1 Þ 0 mod 2: % % % % % If a1 =b1 , then 0th level component of a b is 0 and 1th level component % a % primitive sequence generated a % b is% a1 þ b1 mod 2. Thus a1 þ b1 is %by f% ðxÞ %over % F , so is hðxÞð%a þ%b Þ. Let hðxÞa ¼ ðu ; u ; u ; . . .Þ and 2 1 1 0 0 1 2 % % %
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
577
hðxÞða1 þ b1 Þ ¼ ðs0 ; s1 ; s2 ; . . .Þ over F2 , then by (10), si ¼ 0 if ui ¼ 0. And since%hðxÞð%a1 þ b1 Þ and hðxÞa0 are primitive sequences generated by f ðxÞ over F2 , we get% hðxÞa% 0 hðxÞða1 %þ b1 Þ mod 2, that is a1 þ b1 a0 mod 2. So % % % % % % hðxÞða1 þ b1 ÞhðxÞa0 hðxÞða1 þ b1 Þ mod 2: % % % % % Then, (15) implies ðae3 þ be3 ÞhðxÞa0 hðxÞða1 þ b1 Þ þ ½Ze4 ða0 ; . . . ; ae4 Þ % % % % % % % þ Ze4 ðb0 ; . . . ; be4 ÞhðxÞa0 mod 2: % % % In general, we have ðaei þ bei ÞhðxÞa0 hðxÞða1 þ b1 Þ þ ½Zei1 ða0 ; . . . ; aei1 Þ % % % % % % % þ Zei1 ðb0 ; . . . ; bei1 ÞhðxÞa0 mod 2; % % %
ð16Þ
where i ¼ 2; 3; . . . ; e 2: Finally, by xT 1 2h1 ðxÞ mod f ðxÞ acting on a % and b, similar to (9), we have % ðxT 1Þða2 þ b2 Þ ða1 þ b1 Þh1 ðxÞa0 þ h1 ðxÞða1 þ b1 Þ mod 2; ð17Þ % % % % % % % which implies ðxT 1Þ½ða2 þ b2 ÞhðxÞa0 ða1 þ b1 Þh1 ðxÞa0 hðxÞa0 % % % % % % % þ h1 ðxÞða1 þ b1 ÞhðxÞa0 mod 2: % % %
ð18Þ
On the other hand, by (16) in the case of i ¼ e 2, ða2 þ b2 ÞhðxÞa0 hðxÞða1 þ b1 Þ þ ½Z1 ða0 ; a1 Þ þ Z1 ðb0 ; b1 ÞhðxÞa0 mod 2 % % % % % % % % % % and Z1 ðx0 ; x1 Þ ¼ x1 Z0 ðx0 Þ þ c0 ðx0 Þ; we have ðxT 1Þ½ða2 þ b2 ÞhðxÞa0 % % % T
ðx 1Þ½ðZ1 ða0 ; a1 Þ þ Z1 ðb0 ; b1 ÞÞhðxÞa0 þ hðxÞða1 þ b1 Þ % % % % % % %
ðxT 1Þ½a1 Z0 ða0 Þ þ b1 Z0 ðb0 ÞhðxÞa0 % % % % %
Z1 ða0 ÞhðxÞa0 ðxT 1Þða1 þ b1 Þ % % % %
Z1 ða0 ÞhðxÞa0 ðh1 ðxÞa0 þ h1 ðxÞb0 Þ 0 mod 2: % % % % So (18) implies ða1 þ b1 Þh1 ðxÞa0 hðxÞa0 h1 ðxÞða1 þ b1 ÞhðxÞa0 mod 2: % % % % % % %
ð19Þ
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If a1 =b1 ; then a1 þ b1 a0 mod 2 and the above equation implies % % % % % ð20Þ a0 h1 ðxÞa0 hðxÞa0 h1 ðxÞa0 hðxÞa0 mod 2: % % % % % It is clear that a0 ; h1 ðxÞa0 and hðxÞa0 are linear independent over F2 ; since % 2; h ðxÞc0 % mod 2 and hðxÞc0; 1 mod 2. So, by hðxÞ h1 ðxÞ þ h%1 ðxÞ2 mod 1 Lemma 4, (20) is not true. Thus a1 ¼ b1 and by (19), % by (16) again, we ða2 þ b2 ÞhðxÞa0 0 mod 2, which implies a2 ¼ b%2 : And % % % % % have ðaei þ bei ÞhðxÞa0 0 mod 2, which implies aei ¼ bei ; i ¼ e 3; % % ¼b % % e 4; . .% . ; 2: Finally ae1 e1 since ak ¼ bk ; k ¼ 0; 1; . . . ; e 2; and jða0 ; % % % % % . . . ; ae1 Þ jðb0 ; . . . ; be1 Þ mod 2: Therefore, a ¼ b: % % % % % Secondly, we consider the case e ¼ 3. We have jðx0 ; x1 ; x2 Þ ¼ x2 þ Zðx0 ; x1 Þ and Zðx0 ; x1 Þ ¼ x1 Z0 ðx0 Þ þ c0 ðx0 Þ. Since jða0 ; a1 ; a2 Þ jðb0 ; b1 ; b2 Þ and % % % % % % a0 ¼ b0 , it follows that % % ð21Þ a2 þ b2 ða1 þ b1 ÞZ0 ða0 Þ mod 2: % % % % % By Lemmas 3 and 2, ðxT 1Þða2 þ b2 Þ ðxT 1Þðða1 þ b1 ÞZ0 ða0 ÞÞ % % % % %
Z0 ða0 ÞðxT 1Þða1 þ b1 Þ % % %
Z0 ða0 Þðh1 ðxÞa0 þ h1 ðxÞb0 Þ 0 mod 2: % % % By (17), we have h1 ðxÞða1 þ b1 Þ ða1 þ b1 Þh1 ðxÞa0 mod 2: % % % % % If a1 þ b1 c0 mod 2, then a1 þ b1 mod 2 is a primitive sequence over % 2, % h ðxÞða þ b Þ and h ðxÞa are also F2 ; % and% since h1 ðxÞc0 mod 1 1 1 1 0 % %is in contradiction % primitive sequences over F2 . This condition with the above equation. So a1 þ b1 0 mod 2, and by (21), we have a2 ¼ b2 and % % % % a ¼ b. % Finally, % considering the case e ¼ 2, we have jðx0 ; x1 Þ ¼ x1 þ Zðx0 Þ. So j ða0 ; a1 Þ jðb0 ; b1 Þ mod 2 and a0 ¼ b0 imply a1 ¼ b1 . Hence a ¼ b: & % % % % % % % % % % 3.
COMPRESSION MAPPINGS OVER GALOIS RINGS
Let p be a prime, Zp the p-adic integer ring and Qp the p-adic number field. Let K be an unramified extension of Qp with degree r, R the integer ring of K, then GRðpe ; rÞ ¼ R=p e R is called a Galois ring, where e is a positive integer.
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
579
Remark 2. (1) Let gðxÞ be a monic polynomial over Z=ðp e Þ with degree r. If gðxÞ mod p is irreducible over Fp , then A½x=ðgðxÞÞ ffi GRðpe ; rÞ, where A ¼ Z=ðpe Þ. (2) GRðp e ; 1Þ ¼ Z=ðpe Þ. (3) GRðp e ; rÞ is a local ring with the maximal ideal ðpÞ ¼ pGRðpe ; rÞ ¼ fpa j a 2 GRðp e ; rÞg and GRðp; rÞ ¼ GRðpe ; rÞ=ðpÞ ¼ Fpr is a finite field of p r elements. r (4) Let O ¼ fa 2 GRðpe ; rÞjap ¼ ag, then O consists of pr elements, which are distinct modulo p. So the mapping O ! Fpr , a/a mod p is bijective. Furthermore, each element a in GRðpe ; rÞ may be written uniquely as a ¼ a0 þ a1 p þ þ ae1 pe1 ; where ai 2 O. We call O the p-adic coordinate set of GRðpe ; rÞ (see [5]). (5) Let a be a sequence over GRðpe ; rÞ, then a may be written uniquely as % % a ¼ a0 þ a1 p þ þ ae1 pe1 ; % % % % where ai ¼ ðai0 ; ai1 ; . . .Þ is a sequence over O, i ¼ 0; 1; . . . ; e 1: The sequence% ai is called ith level component of a and ae1 the highest level % % component% of a. % Now we set p ¼ 2 and let f ðxÞ be a monic polynomial over GRð2e ; rÞ. If f ð0Þc0 mod 2, then there exists a positive integer P such that f ðxÞ divides xP 1 and the least such P is called the period of f ðxÞ over GRð2e ; rÞ; denoted by perðf ðxÞÞ. For a monic polynomial f ðxÞ over GRð2e ; rÞ with degree n, the period of f ðxÞ is upper bounded by 2e1 ð2rn 1Þ and f ðxÞ is called a primitive polynomial if perðf ðxÞÞ ¼ 2e1 ð2rn 1Þ. Let f ðxÞ be a primitive polynomial of degree n over GRð2e ; rÞ; then f ðxÞ mod 2 is a primitive polynomial over F2r : Let a be a sequence over GRð2e ; rÞ; generated by a primitive polynomial f ðxÞ of %degree n with ac0 mod 2, then perðaÞ ¼ perðf ðxÞÞ ¼ 2e1 T and perðai Þ ¼ 2i T where T ¼ 2% rn 1, i ¼ 0; 1; . . . ; e % 1. Especially perðae1 Þ ¼ perðaÞ% ¼ 2e1 T . % % Lemma 6. Let f ðxÞ be a primitive polynomial of degree n over GRð2e ; rÞ, T ¼ 2rn 1; then there exists hi ðxÞ over GRð2e ; rÞ of degree less than n, i ¼ 1; 2; . . . ; e 1; such that x2
i1
T
1 2i hi ðxÞ mod f ðxÞ:
ð22Þ
Furthermore, all hi ðxÞc0 mod 2, h2 ðxÞ he1 ðxÞ mod 2 and h2 ðxÞ
h1 ðxÞ þ h1 ðxÞ2 modð2; f ðxÞÞ.
580
WENFENG AND XUANYONG
Definition 1. Let f ðxÞ be a primitive polynomial over GRð2e ; rÞ. If e 3 and degðh2 ðxÞ mod 2Þ 1 or e ¼ 2 and degðh1 ðxÞ mod 2Þ 1, then f ðxÞ is called a strongly primitive polynomial over GRð2e ; rÞ. Lemma 7. Let f ðxÞ be a primitive polynomial over GRð2e ; rÞ, Zðx0 ; x1 ; . . . ; xe2 Þ a function of e 1 variables over F2r and jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ: For a; b 2 Gðf ðxÞ; GRð2e ; rÞÞ; if jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; be1 Þ mod 2; then % % % % a0 ¼%b0%. % % The proof is similar to the one of Lemma 5. Lemma 8. Let O be the p-adic coordinate set of GRð2e ; rÞ, d 2 GRð2e ; rÞ, a; b 2 O; then 1. 2. 3. 4.
r1
ðabÞ2
r
d2 d mod 2. s For s e 1, d2 2 O. ab 2 O. rs Let s be a positive integer such that rs e 1; then ða þ bÞ2 2 O, 2 O and rs
r1
a þ b ða þ bÞ2 þ 2ðabÞ2
mod 22 : r
Proof. (1) Let d ¼ l þ 2x; where l 2 O, x 2 GRð2e ; rÞ. Since l2 ¼ l, d d mod 2. (2) If d 0 mod 2; then d ¼ 2x; where x 2 GRð2e ; rÞ. Since 2s 2e1 e; s we have d2 ¼ 0 2 O: If dc0 mod 2; then d ¼ l þ 2x; where l 2 O, s s s s s x 2 GRð2e ; rÞ. So d2 ¼ ðl þ 2xÞ2 ¼ l2 þ 21þs l2 1 x þ ¼ l2 2 O: r 2r 2r (3) ðabÞ ¼ a2 b ¼ ab; so ab 2 O. rs r1 (4) By (2) and (3), ða þ bÞ2 2 O; ðabÞ2 2 O: Since 2r
k
k
k
k1
k1
ða þ bÞ2 a2 þ b2 þ 2a2 b2
mod 22 ;
we have rs
rs
rs
ða þ bÞ2 a2 þ b2 þ 2a2
a þ b þ 2ða2
rs1
rðs1Þ
rs
r1
So a þ b ða þ bÞ2 þ 2ðabÞ2
mod 22 :
&
r1
rs1
Þ2 ðb2
r1
a þ b þ 2ðabÞ2
b2
rðs1Þ
mod 22 :
Þ2
r1
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
581
Lemma 9. Let f ðxÞ be a primitive polynomial of degree n over the finite field F2r , u ¼ ðu0 ; u1 ; . . .Þ, v ¼ ðv0 ; v1 ; . . .Þ 2 Gðf ðxÞ; F2r Þ and v=0. If vi ¼ 0 % 0, then there exists % % implies ui ¼ c 2 F2r such that u ¼ c v. % % Proof. If u ¼ 0, then we set c ¼ 0 and get u ¼ cv. Now assume u=0, % condition. Since f ðxÞ is a primitive % polynomial % % n then v=0 by the of degree % and u and v are generated by f ðxÞ, there exists a nonnegative integer k such that % ðuk ; %ukþ1 ; . . . ; ukþn1 Þ ¼ ð0; 0; . . . ; 0; aÞ and ðvk ; vkþ1 ; . . . ; vkþn1 Þ ¼ ð0; 0; . . . ; 0; bÞ, where a=0 and b=0: Then it is clear that u ¼ cv, where % % c ¼ ab1 : & Lemma 10. Let f ðxÞ be a primitive polynomial of degree n over GRð2e ; rÞ, T ¼ 2rn 1, a ¼ a0 þ a1 2 þ þ ae1 2e1 2 Gðf ðxÞ; GRð2e ; rÞÞ and a0 =0; % % Þ ¼ 2e1 T ¼ perðaÞ ¼ perðf ðxÞÞ. % then perðai Þ ¼%2i T ,% especially perðae1 % % % The proof is similar to one of Lemma 1 cited from [1]. Lemma 11. Let f ðxÞ be a strongly primitive polynomial over GRð2e ; rÞ, e 3, a; b 2 Gðf ðxÞ; GRð2e ; rÞÞ and a0 ¼ b0 =0. If there exists c 2 F2r such % b% ca mod 2 and % % that a1 þ 1 0 % % % a0 h1 ðxÞa0 ðh2 ðxÞa0 Þ2 cðh1 ðxÞða0 ÞÞ2 ðh2 ðxÞa0 Þ2 mod 2; ð23Þ % % % % % where h1 ðxÞ and h2 ðxÞ is defined by (22), then a1 ¼ b1 ; i.e. c ¼ 0: % % Proof. Assume a1 =b1 ; that is, c=0. If a0 , h1 ðxÞa0 and h% 2 ðxÞ%a0 are linear independent over F2r , then, by Lemma 4, % % % a0 h1 ðxÞa0 ðh2 ðxÞa0 Þ2 ccðh1 ðxÞða0 ÞÞ2 ðh2 ðxÞa0 Þ2 mod 2; % % % % % that is a contradiction. Now suppose that a0 , h1 ðxÞa0 and h2 ðxÞa0 are linear % % % over F r , dependent over F2r : Since a0 is an m-sequence generated by f ðxÞ 2 % deg ðh1 ðxÞ mod 2Þ 1, deg ðh2 ðxÞ mod 2Þ 1 and h1 ðxÞch2 ðxÞ mod 2, we conclude that any two of a0 , h1 ðxÞa0 and h2 ðxÞa0 are linear independent % a ¼ c% h ðxÞa þ c %h ðxÞa over F r , where c over F2r . Thus, we can assume 0 1 1 0 2 2 0 2 1 % in F r . % We write % a ¼ ða ; a ; a ; . . .Þ; and c2 are nonzero elements 2 0 0 1 2 c1 h1 ðxÞa0 ¼ ðb0 ; b1 ; b2 ; . . .Þ; c2 h2 ðxÞa0 ¼ ðg0 ; g1 ; g2 ; . . .Þ %over F2r . Since c1 h1 ðxÞa% 0 and c2 h2 ðxÞa0 are linear% independent over F2r ; there exists % nonnegative integer % t such that bt ¼ gt =0: Then yt =0 in 2 cðh1 ðxÞða0 ÞÞ ðh2 ðxÞa0 Þ2 ¼ ðy0 ; y1 ; y2 ; . . .Þ. But at ¼ bt þ gt ¼ 2bt ¼ 0, by (23), % we get a% contradiction. & Lemma 12. Let gðxÞ ¼ cn xn þ cn1 xn1 þ þ c0 2 F2r ½x, f ðxÞ ¼ c2n xn þ þ þ c20 and u a sequence over F2r . Then ðgðxÞuÞ2 ¼ f ðxÞu2 : % Especially, ððxk 1ÞuÞ2 ¼ %ðxk 1Þðu2 Þ for any positive integer k.% % % The proof is easy.
c2n1 xn1
582
WENFENG AND XUANYONG
Theorem 3. Let f ðxÞ be a strongly primitive polynomial of degree n over GRð2e ; rÞ, jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ; where Zðx0 ; x1 ; . . . ; xe2 Þ is a function of e 1 variables over F2r . Set Ze2 ðx0 ; x1 ; . . . ; xe2 Þ ¼ Zðx0 ; x1 ; . . . ; xe2 Þ: If Ze2 ðx0 ; x1 ; . . . ; xe2 Þ satisfies Ze2 ðx0 ; x1 ; . . . ; xe2 Þ ¼ xe2 Ze3 ðx0 ; x1 ; . . . ; xe3 Þ þ ce3 ðx0 ; x1 ; . . . ; xe3 Þ and Zi ðx0 ; x1 ; . . . ; xi Þ ¼ xi Zi1 ðx0 ; x1 ; . . . ; xi1 Þ þ ci1 ðx0 ; x1 ; . . . ; xi1 Þ; where i ¼ 1; 2; . . . ; e 2, Zi1 ðx0 ; x1 ; . . . ; xi1 Þ and ci1 ðx0 ; x1 ; . . . ; xi1 Þ are functions of i variables over F2r , then the compression mapping ( Gðf ðxÞ; GRð2e ; rÞÞ ! F21 r ; F: a ¼ a0 þ a1 2 þ þ ae1 2e1 /jða0 ; a1 ; . . . ; ae1 Þ mod 2 % % % % % % % is injective, that is, for a; b 2 Gðf ðxÞ; GRð2e ; rÞÞ, a ¼ b if and only if % Þ mod 2. % % jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; b%e1 % % % % Proof. Let a; b 2 Gðf ðxÞ; GRð2e ; rÞÞ satisfying % % jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; be1 Þ mod 2: % % % % By Lemma 7, a0 ¼ b0 . It is not harmful to assume a0 =0. % % ðxÞ mod 2; (1) Assume e% 4. Since h2 ðxÞ he1 hðxÞ ¼ hi ðxÞ mod 2 over F2r , 2 i e 1. Since
we
ae1 þ ae2 Ze3 ða0 ; . . . ; ae3 Þ þ ce3 ða0 ; . . . ; ae3 Þ % % % % % %
be1 þ be2 Ze3 ðb0 ; . . . ; be3 Þ þ ce3 ðb0 ; . . . ; be3 Þ mod 2 % % % % % %
set
ð24Þ
and by Lemma 10 the periods of ce3 ða0 ; . . . ; ae3 Þ and ce3 ðb0 ; . . . ; be3 Þ e3 % % divide 2e3 T , where T ¼ 2rn 1, by x2 T% 1 acting on (24), we% can get ðx2
e3
T
1Þðae1 þ ae2 Ze3 ða0 ; . . . ; ae3 ÞÞ % % % %
ðx2 T 1Þðbe1 þ be2 Ze3 ðb0 ; . . . ; be3 ÞÞ mod 2: % % % % e3
ð25Þ
Furthermore, by Lemma 3, we have e3
ðx2
e3
1Þae1 þ Ze3 ða0 ; . . . ; ae3 Þðx2 T 1Þae2 % % % % 2e3 T 2e3 T
ðx 1Þbe1 þ Ze3 ðb0 ; . . . ; be3 Þðx 1Þbe2 mod 2: % % % % T
ð26Þ
583
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
Since a0 ¼ b0 and by Lemma 6 x2 % that % ðx2
e3
T
e3
T
1 2e2 he2 ðxÞ mod f ðxÞ, it follows
1Þae2 hðxÞa0 hðxÞb0 ðx2 % % %
e3
T
1Þbe2 mod 2: %
So (26) implies that ðx2
e3
T
1Þðae1 þ be1 Þ % %
½Ze3 ða0 ; . . . ; ae3 Þ þ Ze3 ðb0 ; . . . ; be3 ÞhðxÞa0 mod 2: % % % % %
ð27Þ
e3
By x2 T 1 2e2 he2 ðxÞ mod f ðxÞ acting on a ¼ a0 þ a1 2 þ þ % % % ae1 2e1 , % e3 ðx2 T 1Þðae2 þ ae1 2Þ he2 ðxÞða0 þ a1 2Þ mod 22 : ð28Þ % % % % Let he2 ðxÞða0 þ a1 2Þ u þ v2 mod 22 ; % % % %
ð29Þ
where u and v are 0th and 1th level components of he2 ðxÞða0 þ a1 2Þ, % % % respectively. It %is clear that u he2 ðxÞa0 mod 2. So, by (28) % % e3 e3 x2 T ae2 þ ðx2 T 1Þae1 2 ae2 þ u þ v2 mod 22 : ð30Þ % % % % rs
r1
By Lemma 8, ae2 þ u ðae2 þ uÞ2 þ 2ðae2 uÞ2 mod 22 , where s is a % rs % e 1. % Since ða% %þ uÞ2rs is a sequence over O; % positive integer such that e2 % % by (30), it follows that ðx2
e3
r1
T
1Þae1 v þ ðuae2 Þ2 % % %%
T
1Þbe1 w þ ðube2 Þ2 % % %%
mod 2:
ð31Þ
mod 2;
ð32Þ
Similarly, we have ðx2
e3
r1
where w is determined by % he2 ðxÞðb0 þ b1 2Þ u þ w2 mod 22 : % % % % Since a0 ¼ b0 ; by (29) and (33), it follows % 2, and by (31) and (32), we have hðxÞða1 þ b%1 Þ mod % % e3
ðx2
T
r1
r1
1Þðae1 þ be1 Þ u2 ðae2 þ be2 Þ2 % % % % %
ð33Þ that
vþw¼ % %
þ hðxÞða1 þ b1 Þ mod 2: % %
ð34Þ
584
WENFENG AND XUANYONG
By (27) and u hðxÞa0 mod 2, % % r1
u2 ðae2 þ be2 Þ2 % % %
r1
½Ze3 ða0 ; . . . ; ae3 Þ % % þ Ze3 ðb0 ; . . . ; be3 Þu þ hðxÞða1 þ b1 Þ mod 2: % % % % %
ð35Þ
Then r1
r1
uðae2 þ be2 Þ ½u2 ðae2 þ be2 Þ2 2 % % % % % %
½Ze3 ða0 ; . . . ; ae3 Þ þ Ze3 ðb0 ; . . . ; be3 Þ2 u2 % % % % % þ ½hðxÞða1 þ b1 Þ2 mod 2: % % If e 5, x2 Then
e4
T
ð36Þ
1 2e3 he3 ðxÞ mod f ðxÞ acts on a and b continuously. % %
e4
T
1Þðae3 þ ae2 2Þ he3 ðxÞða0 þ a1 2Þ mod 22 ; % % % %
e4
T
1Þðbe3 þ be2 2Þ he3 ðxÞðb0 þ b1 2Þ mod 22 : % % % %
ðx2 ðx2
Similar to (34), we have ðx2
e4
T
r1
r1
1Þðae2 þ be2 Þ u2 ðae3 þ be3 Þ2 % % % % %
þ hðxÞða1 þ b1 Þ mod 2 % % ð37Þ
and so ½ðx2
e4
T
r1
r1
1Þðae2 þ be2 Þu u½u2 ðae3 þ be3 Þ2 % % % % % % %
þ hðxÞða1 þ b1 Þ mod 2: % %
Since perðuÞ ¼ T ; we have % ðx2
e4
T
1Þ½ðae2 þ be2 Þu % % % r1
uu2 ðae3 þ be3 Þ2 % % %%
r1
þ uhðxÞða1 þ b1 Þ mod 2: % % %
ð38Þ
585
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
And by (36), it follows that ðae2 þ be2 Þu % % %
u2 ½ae3 Ze4 ða0 ; . . . ; ae4 Þ þ be3 Ze4 ðb0 ; . . . ; be4 Þ % % % % % % % þ ce4 ða0 ; . . . ; ae4 Þ þ ce4 ðb0 ; . . . ; be4 Þ2 þ ½hðxÞða1 þ b1 Þ2 mod 2: % % % % % % The periods of ðce4 ða0 ; . . . ; ae4 Þ þ ce4 ðb0 ; . . . ; be4 ÞÞ2 u2 and ðhðxÞða1 þ % % % % b1 ÞÞ2 divide 2e4 T , so it%follows% that % e4 ðx2 T 1Þ½ðae2 þ be2 Þu % % %
ðx2
e4
T
1Þ½ae3 Ze4 ða0 ; . . . ; ae4 Þ þ be3 Ze4 ðb0 ; . . . ; be4 Þ2 u2 mod 2: % % % % % % %
Comparing with (38) and by Lemma 12 we get e4
½Ze4 ða0 ; . . . ; ae4 Þðx2 % % r1
T
u u2 ðae3 þ be3 Þ2 % % % % Since ðx2
e4
T
e4
1Þae3 þ Ze4 ðb0 ; . . . ; be4 Þðx2 % % % r1
T
1Þbe3 2 u2 % %
þ u hðxÞða1 þ b1 Þ mod 2: % % %
1Þae3 he3 ðxÞa0 u mod 2, then % % % ½Ze4 ða0 ; . . . ; ae4 Þ þ Ze4 ðb0 ; . . . ; be4 Þ2 u4 % % % % % r1
u u2 ðae3 þ be3 Þ2 % % % %
r1
þ u hðxÞða1 þ b1 Þ mod 2: % % %
So ½Ze4 ða0 ; . . . ; ae4 Þ þ Ze4 ðb0 ; . . . ; be4 Þ4 u8 % % % % % r1
r1
½u u2 ðae3 þ be3 Þ2 % % % %
þ u hðxÞða1 þ b1 Þ2 % % %
u2 uðae3 þ be3 Þ þ u2 ½hðxÞða1 þ b1 Þ2 mod 2; % % % % % % % that is, uðae3 þ be3 Þu2 ½Ze4 ða0 ; . . . ; ae4 Þ þ Ze4 ðb0 ; . . . ; be4 Þ4 u8 % % % % % % % % % 2 2 þ u ½hðxÞða1 þ b1 Þ mod 2: % % %
ð39Þ
Let u ðu0 ; u1 ; . . .Þ mod 2; hðxÞða1 þ b1 Þ ¼ ðw0 ; w1 ; . . .Þ mod 2; by (36), it % by% Lemma 9, hðxÞða þ b Þ cu mod 2 follows%that if ui ¼ 0; then wi 0: So 1 1 % % %
586
WENFENG AND XUANYONG
for some c 2 F2r ; and (39) implies uðae3 þ be3 Þ u6 ½Ze4 ða0 ; . . . ; ae4 Þ þ Ze4 ðb0 ; . . . ; be4 Þ4 % % % % % % % % þ ½hðxÞða1 þ b1 Þ2 mod 2: % % From the above discussion, we deduce the following formula: uðaei þ bei Þ uki ðZei1 ða0 ; . . . ; aei1 Þ þ Zei1 ðb0 ; . . . ; bei1 ÞÞ2 % % % % % % % % þ ½hðxÞða1 þ b1 Þ2 mod 2; % %
i1
ð40Þ
where ki is a positive integer, i ¼ 2; 3; . . . ; e 2. Take i ¼ e 2; then uða2 þ b2 Þ uke2 ½Z1 ða0 ; a1 Þ þ Z1 ðb0 ; b1 Þ2 % % % % % % % %
e3
þ ½hðxÞða1 þ b1 Þ2 mod 2: % %
ð41Þ
Finally, xT 1 2h1 ðxÞ mod f ðxÞ acts on a and b. Similar to (34), we have % % r1
r1
ðxT 1Þða2 þ b2 Þ ðh1 ðxÞa0 Þ2 ða1 þ b1 Þ2 % % % % %
þ h1 ðxÞða1 þ b1 Þ mod 2; % %
ð42Þ
which deduces r1
r1
ðxT 1Þða2 þ b2 Þu ðh1 ðxÞa0 Þ2 ða1 þ b1 Þ2 u þ h1 ðxÞða1 þ b1 Þu mod 2:ð43Þ % % % % % % % % % % Since Z1 ðx0 ; x1 Þ ¼ x1 Z0 ðx0 Þ þ c0 ðx0 Þ and hðxÞða1 þ b1 Þ cu mod 2, (41) im% % % plies e3
ðxT 1Þða1 Z0 ða0 Þ þ b1 Z0 ðb0 ÞÞ2 uke2 % % % % % 2r1 2r1
ðh1 ðxÞa0 Þ ða1 þ b1 Þ u þ h1 ðxÞða1 þ b1 Þu mod 2: % % % % % % %
ð44Þ
Since ðxT 1Þa1 ðxT 1Þb1 h1 ðxÞa0 mod 2 and Z0 ða0 Þ Z0 ðb0 Þ mod 2; we % % % % % have e3
ðxT 1Þða1 Z0 ða0 Þ þ b1 Z0 ðb0 ÞÞ2 uke2 % % % % % e3
½ðxT 1Þa1 Z0 ða0 Þ þ ðxT 1Þb1 Z0 ðb0 Þ2 uke2 0 mod 2: % % % % %
587
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
So (44) implies r1
r1
ðh1 ðxÞa0 Þ2 ða1 þ b1 Þ2 u þ h1 ðxÞða1 þ b1 Þu 0 mod 2 % % % % % % % and then r1
r1
ðh1 ðxÞða1 þ b1 ÞÞ2 u2 ½ðh1 ðxÞa0 Þ2 ða1 þ b1 Þ2 u2 % % % % % % %
h1 ðxÞa0 ða1 þ b1 Þu2 mod 2: % % % %
ð45Þ
Because hðxÞða1 þ b1 Þ cu mod 2 and h2 ðxÞa0 u mod 2, hðxÞða1 þ b1 Þ
% % % % that chðxÞa0 hðxÞðca%0 Þ; which implies a1 þ b1 ca0% mod% 2. And (45) implies % % % % % a0 h1 ðxÞa0 ðh2 ðxÞa0 Þ2 cðh1 ðxÞða0 ÞÞ2 ðh2 ðxÞa0 Þ2 mod 2: % % % % % By Lemma 11, we get a1 ¼ b1 . Thus uða2 þ b2 Þ 0 mod 2 by (41). Since % that % a þ b %is% a primitive % a0 ¼ b0 ; a1 ¼ b1 , it follows sequence over F2r or 2 2 %zero sequence. % % %And since u is a primitive % % sequence over F r ; we get a þ b
2 2 2 % by (40), it follows that a ¼ b ; j ¼ 3; .%. . ; e % 2: 0 mod 2; that is, a2 ¼ b2 . So j j % % % Þ mod 2, we have Finally, by the condition jða0 ; . . . ; ae1 Þ jðb0 ; . . %. ; be1 % % % % ae1 ¼ be1 ; and then a ¼ b. % (2) % Assume e%¼ 3;% then jðx0 ; x1 ; x2 Þ ¼ x2 þ Zðx0 ; x1 Þ and Zðx0 ; x1 Þ ¼ x1 Z0 ðx0 Þ þ c0 ðx0 Þ. Since jða0 ; a1 ; a2 Þ jðb0 ; b1 ; b2 Þ and a0 ¼ b0 , % % % % % % % % it follows that a2 þ b2 ða1 þ b1 ÞZ0 ða0 Þ mod 2: % % % % %
ð46Þ
By Lemmas 3 and 6, ðxT 1Þða2 þ b2 Þ ðxT 1Þðða1 þ b1 ÞZ0 ða0 ÞÞ % % % % %
Z0 ða0 ÞðxT 1Þða1 þ b1 Þ % % %
Z0 ða0 Þðh1 ðxÞa0 þ h1 ðxÞb0 Þ % % %
0 mod 2: r1
r1
By (42), we have h1 ðxÞða1 þ b1 Þ ða1 þ b1 Þ2 ðh1 ðxÞa0 Þ2 mod 2, and so % % % % % r1 r1 ðh1 ðxÞða1 þ b1 ÞÞ2 ½ða1 þ b1 Þ2 ðh1 ðxÞa0 Þ2 2 % % % % %
ða1 þ b1 Þh1 ðxÞa0 mod 2: % % % If a1 þ b1 c0 mod 2, that is, a1 þ b1 mod 2 and h1 ðxÞða1 þ b1 Þ are % % sequences over F %r . %Since degðh ðxÞ mod% 2Þ % 1; it primitive 2 1
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follows that and a1 þ b1 0 mod 2; Furthermore, by (46), a2 ¼ b2 % % % % and a ¼ b. % % (3) Assume e ¼ 2; then jðx0 ; x1 Þ ¼ x1 þ Zðx0 Þ. So jða0 ; a1 Þ jðb0 ; b1 Þ mod % % % % 2 and a0 ¼ b0 imply a1 ¼ b1 . Hence a ¼ b: &. % % % % % % Remark 3. Theorems 2 and 3 show that the binary sequence jða0 ; . . . ; % for ae1 Þ contains all information of the original sequence a. We guess that % any Z, Theorem 3 is also correct. REFERENCES 1. Z. D. Dai, Binary sequences derived from ML-sequences over rings I. Periods and minimal polynomials, J. Cryptol. 5, No. 4 (1992), 193–207. 2. Z. D. Dai, T. Beth, and D. Gollman, Lower bounds for the linear complexity of sequences over residue rings, in ‘‘Advances in Cryptology}EUROCRYPT’90’’ (I. B. Damgard Ed.), Lecture Notes in Computer Science, Vol. 473, pp. 189–195, Springer-Verlag, Berlin, 1991. 3. M. Q. Huang, ‘‘Analysis and Cryptologic Evaluation of Primitive Sequences over an Integer Residue Ring,’’ Doctoral dissertation of Graduate School of USTC, Academia Sinica, 1988. 4. M. Q. Huang and Z. D. Dai, Projective maps of linear recurring sequences with maximal p-adic periods, Fibonacci Quart. 30, No. 2 (1992), 139–143. 5. V. L. Kurakin, The first coordinate sequence of a linear recurrence of maximal period over a Galois ring, Discrete Math. Appl. 4, No. 2 (1994), 129–141. 6. A. S. Kuzmin and A. A. Nechaev, ‘‘A Construction of Noise Stable Codes using Linear Recurrences Over Galois Rings,’’ Russian Mathematical Survey, Vol. 47, pp. 189–190, 1992. 7. A. S. Kuzmin and A. A. Nechaev, ‘‘Linear Recurring Sequences over Galois Rings,’’ Russian Mathematical Survey, Vol. 48, pp. 171–172, 1993. 8. Qi Wenfeng, Yang Junhui, and Zhou Jinjun, ML-sequences over rings Z=ð2e Þ, in ‘‘Advances in Cryptology}ASIACRYPT’98,’’ (K. Ohta and D. Pei, Eds.), Lecture Notes in Computer Science, Vol. 1514, pp. 315–325, Springer-Verlag, Berlin, Heidelberg, 1998.
Compressing Mappings on Primitive Sequences over Z=ð2e Þ and Its Galois Extension1 Qi Wenfeng2 and Zhu Xuanyong Department of Applied Mathematics, Zhengzhou Information Engineering University, P.O. Box 1001-745, Zhengzhou, 450002, People’s Republic of China E-mail: [email protected],[email protected] Communicated by Rudolf Lidl Received March 11, 2001; revised February 10, 2002; published online June 25, 2002
Let f ðxÞ be a strongly primitive polynomial of degree n over Z=ð2e Þ, Zðx0 ; x1 ; . . . ; xe2 Þ a Boolean function of e 1 variables and jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ Gðf ðxÞ; Z=ð2e ÞÞ denotes the set of all sequences over Z=ð2e Þ generated by f ðxÞ, F21 the set of all sequences over the binary field F2 , then the compressing mapping ( F:
Gðf ðxÞ; Z=ð2e ÞÞ ! F21 ; a ¼ a0 þ a1 2 þ þ ae1 2e1 /jða0 ; a1 ; . . . ; ae1 Þ mod 2 % % % % % % %
is injective, that is, for a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ, a ¼ b if and only if FðaÞ ¼ FðbÞ, i.e., % part of the paper, % we general% jða0 ; . . . ; ae1 Þ ¼ jðb0 ; . .%. ;%be1 Þ mod 2. In the %second % % Galois rings. # 2002 Elsevier Science (USA) ize% the above result %over the Key Words: primitive polynomial; Galois ring; linear sequence; compressing mapping.
1.
INTRODUCTION
Let R be a ring, f ðxÞ ¼ xn þ cn1 xn1 þ þ c0 a monic polynomial over R, the sequence a ¼ ða0 ; a1 ; a2 ; . . .Þ over R satisfying the recursion % aiþn ¼ ðc0 ai þ c1 aiþ1 þ þ cn1 aiþn1 Þ; i ¼ 0; 1; 2; . . . 1 The work is supported by HAIPURT and the Special Fund of National Excellently Doctoral Paper. 2 To whom correspondence should be addressed.
570 1071-5797/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
571
is called a linear recurring sequence of degree n over R, generated by f ðxÞ. We will use the notation Gðf ðxÞ; RÞ for the set of all sequences over R generated by f ðxÞ. Let a ¼ ða0 ; a1 ; a2 ; . . .Þ and b ¼ ðb0 ; b1 ; b2 ; . . .Þ be sequences over R and % % c 2 R; define a þ b ¼ ða0 þ b0 ; a1 þ b1 ; a2 þ b2 ; . . .Þ; % % ca ¼ ðca0 ; ca1 ; ca2 ; . . .Þ; % ab ¼ ða0 b0 ; a1 b1 ; a2 b2 ; . . .Þ %% and the shift operator x of sequence as xa ¼ ða1 ; a2 ; a3 ; . . .Þ and xk a ¼ ðak ; akþ1 ; akþ2 ; . . .Þ for k ¼ 0; 1; 2; . . . : Then for% any polynomial f ðxÞ over% R, a 2 Gðf ðxÞ; RÞ if and only if f ðxÞa ¼ 0. % Let f ðxÞ be a monic polynomial % of degree n over Z=ð2e Þ with f ð0Þc0 mod 2, then there exists a positive integer P such that f ðxÞ divides xP 1 over Z=ð2e Þ. The least such P is called the period of f ðxÞ over Z=ð2e Þ and denoted by perðf ðxÞÞ. The period of f ðxÞ is upper bounded by 2e1 ð2n 1Þ, where n ¼ deg f ðxÞ. A monic polynomial f ðxÞ of degree n over Z=ð2e Þ is called a primitive polynomial if perðf ðxÞÞ ¼ 2e1 ð2n 1Þ. Let f ðxÞ be a primitive polynomial of degree n over Z=ð2e Þ, T ¼ 2n 1, then f ðxÞ mod 2 is a primitive polynomial over the binary field F2 and for i ¼ 1; 2; . . . ; e 1, we have x2
i1
T
1 2i hi ðxÞ mod f ðxÞ;
ð1Þ
where hi ðxÞ is a polynomial over Z=ð2e Þ of degree less than n such that hi ðxÞc0 mod 2. Furthermore, h2 ðxÞ he1 ðxÞ mod 2; h2 ðxÞ h1 ðxÞ þ h1 ðxÞ2 mod ð2; f ðxÞÞ:
ð2Þ
If e 3 and h2 ðxÞc1 mod 2 or e ¼ 2 and h1 ðxÞc1 mod 2; then f ðxÞ is called a strongly primitive polynomial over Z=ð2e Þ (see [1, 3]). Any element a in Z=ð2e Þ has a unique binary decomposition as a ¼ a0 þ a1 2 þ þ ae1 2e1 , ai 2 f0; 1g. Similarly, a sequence a over Z=ð2e Þ has a unique binary decomposition as a ¼ a0 þ a1 2 þ þ ae1% 2e1 , where % over % %f0; 1g. The % sequence a is ai ¼ ðai0 ; ai1 ; ai2 ; . . .Þ is a binary sequence i %called ith level component of a, and a % e1 the highest level component of a. % % properties of the level component of % There are many papers to discuss the a, please refer to [1–5, 7, 8]. %
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In the first part of the paper, we prove the following result. Let f ðxÞ be a strongly primitive polynomial of degree n over Z=ð2e Þ, Zðx0 ; x1 ; . . . ; xe2 Þ a Boolean function of e 1 variables and jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ: F21 denotes the set of all sequences over the binary field F2 ; then the compressing map ( Gðf ðxÞ; Z=ð2e ÞÞ ! F21 ; F: a ¼ a0 þ a1 2 þ þ ae1 2e1 /jða0 ; a1 ; . . . ; ae1 Þ mod 2 % % % % % % % is injective, that is, for a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ, a ¼ b if and only if % Þ% mod 2. % % jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; be1 %In the second % % % part of the paper, we generalize the above result to the one over Galois rings.
2. INJECTIVENESS OF COMPRESSION MAPPINGS OVER Z=ð2e Þ Huang [3] and Huang and Dai [4] proposed the following injectiveness theorem. Theorem 1. mapping
Let f ðxÞ be a primitive polynomial over Z=ð2e Þ, then the ( F:
Gðf ðxÞ; Z=ð2e ÞÞ ! F21 ; a ¼ a0 þ a1 2 þ þ ae1 2e1 /ae1 % % % % %
is injective, that is, for a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ; a ¼ b if and only if ae1 ¼ be1 . % % % % % % Remark 1. Theorem 1 implies that ae1 contains all information of the % original sequence a. % Lemma 1 (Dai [1]). Let f ðxÞ be a primitive polynomial of degree n over Z=ð2e Þ, T ¼ 2n 1, a ¼ a0 þ a1 2 þ þ ae1 2e1 2 Gðf ðxÞ; Z=ð2e ÞÞ and % % perða Þ% ¼ 2e1 T ¼ perðaÞ ¼ perðf ðxÞÞ: a0 =0, then perðai Þ ¼ 2i T% , especially e1 % % % % Lemma 2 (Dai [1]). Let f ðxÞ be a primitive polynomial of degree n over Z=ð2e Þ, T ¼ 2n 1; a ¼ a0 þ a1 2 þ þ ae1 2e1 2 Gðf ðxÞ; Z=ð2e ÞÞ, then i1 % %2 for% i ¼ 1; 2; . . .%; e 1. That is, by (2), ðx2 T 1Þai hi ðxÞa0 mod % % 8 if i ¼ 1; < h1 ðxÞa0 mod 2 i1 % ðx2 T 1Þai
: h2 ðxÞa0 mod 2 if i 2: % %
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
573
Lemma 3. Let S be a positive integer, u a sequence over the ring R with its period dividing S, then for any sequence v %over R, ðxS 1Þðv uÞ ¼ uðxS 1Þv: % %% % % The proof is easy. Lemma 4. Let f ðxÞ be a primitive polynomial of degree n over the finite field F2r ; and a ¼ ða0 ; a1 ; . . .Þ, b ¼ ðb0 ; b1 ; . . .Þ, c ¼ ðc0 ; c1 ; . . .Þ 2 Gðf ðxÞ; F2r Þ. % linear independent % If a, b and c are over F2r , %then the number of zeros in % % % fa0 b0 ; a1 b1 ; . . . ; aT 1 bT 1 g is smaller than the one in fa0 b0 c0 ; a1 b1 c1 ; . . . ; aT 1 bT 1 cT 1 g; where T ¼ 2nr 1. Thus a b=a b c: %% %%% The proof is easy. Lemma 5. Let f ðxÞ be a primitive polynomial of degree n over Z=ð2e Þ, Zðx0 ; x1 ; . . . ; xe2 Þ a Boolean function of e 1 variables and jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ: If jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; be1 Þ mod 2 for a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ; then % % % % % a0 ¼ b%0 . % % Proof. Set T ¼ 2n 1. The periods of Zða0 ; . . . ; ae2 Þ and Zðb0 ; . . . ; be2 Þ e2 % a ; . . .%; a Þ jðb% ; . . . ; b% Þ divide 2e2 T , so, by x2 T 1 acting on jð 0 e1 0 e1 % % % % mod 2, we get ðx2
e2
T
1Þae1 ðx2 %
e2
T
1Þbe1 mod 2: %
Since perðai Þ ¼ 2i T for 0 i e 2, by x2e2T 1 2e1 he1 ðxÞ mod f ðxÞ acting on %a and b, respectively, we have % % e2 ðx2 T 1Þae1 2e1 ¼ 2e1 he1 ðxÞa0 ; % % ðx2
e2
that is,
T
1Þbe1 2e1 ¼ 2e1 he1 ðxÞb0 ; % %
ðx2
e2
T
1Þae1 he1 ðxÞa0 mod 2; % %
ðx2
e2
T
1Þbe1 he1 ðxÞb0 mod 2: % %
So he1 ðxÞa0 he1 ðxÞb0 mod 2: Since he1 ðxÞc0 modð2; f ðxÞÞ; we have % a0 ¼ b0 : &% % % Theorem 2. Let f ðxÞ be a strongly primitive polynomial of degree n over Z=ð2e Þ, Zðx0 ; x1 ; . . . ; xe2 Þ a Boolean function of e 1 variables and jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ;
574
WENFENG AND XUANYONG
then the compressing map ( F:
Gðf ðxÞ; Z=ð2e ÞÞ ! F21 ; a ¼ a0 þ a1 2 þ þ ae1 2e1 /jða0 ; a1 ; . . . ; ae1 Þ mod 2 % % % % % % %
is injective, that is, for a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ, a ¼ b if and only if jða0 ; . . . ; % % 2: % ae1 Þ jðb0 ; . . . ; be1 Þ mod % % % Proof. Let a; b 2 Gðf ðxÞ; Z=ð2e ÞÞ with jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; be1 Þ % 2, we% set mod 2 we shall% %prove a ¼ b. Since h2 ðxÞ % %he1 ðxÞ mod % % hðxÞ ¼ h2 ðxÞ mod 2 which is a polynomial over the field F2 : Since a0 ¼ b0 by Lemma 5, it suffices to consider the case a0 ¼ b0 =0. Set % % Ze2 ðx0 ; x%1 ; . . . %; xe2 Þ ¼ Zðx0 ; x1 ; . . . ; xe2 Þ, then Ze2 ðx0 ; x1 ; . . . ; xe2 Þ ¼ xe2 Ze3 ðx0 ; x1 ; . . . ; xe3 Þ þ ce3 ðx0 ; x1 ; . . . ; xe3 Þ: In general, we have Zi ðx0 ; x1 ; . . . ; xi Þ ¼ xi Zi1 ðx0 ; x1 ; . . . ; xi1 Þ þ ci1 ðx0 ; x1 ; . . . ; xi1 Þ;
ð3Þ
where Zi1 ðx0 ; x1 ; . . . ; xi1 Þ and ci1 ðx0 ; x1 ; . . . ; xi1 Þ are Boolean functions with i variables, i ¼ 1; 2; . . . ; e 1: Firstly, we consider the case e 4: Since ae1 þ ae2 Ze3 ða0 ; . . . ; ae3 Þ þ ce3 ða0 ; . . . ; ae3 Þ % % % % % %
be1 þ be2 Ze3 ðb0 ; . . . ; be3 Þ þ ce3 ðb0 ; . . . ; be3 Þ mod 2 % % % % % % and the periods of ce3 ða0 ; . . . ; ae3 Þ and ce3 ðb0 ; . . . ; be3 Þ divide 2e3 T , it % % % % follows ðx2
e3
ðx2 and
ðx2
e3
T
e3
T
1Þce3 ða0 ; . . . ; ae3 Þ 0 mod 2; % % 1Þce3 ðb0 ; . . . ; be3 Þ 0 mod 2 % %
T
1Þðae1 þ ae2 Ze3 ða0 ; . . . ; ae3 ÞÞ % % % % 2 T
ðx 1Þðbe1 þ be2 Ze3 ðb0 ; . . . ; be3 ÞÞ mod 2: % % % % e3
ð4Þ
The periods of Ze3 ða0 ; a1 ; . . . ; ae3 Þ and Ze3 ðb0 ; b1 ; . . . ; be3 Þ divide 2e3 T e3 % hðxÞ % a %hðxÞb ðx2e3 T% %1Þb %mod 2, so (4) and ðx2 T 1Þae2
0 0 e2 % % % %
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
575
implies ðx2
e3
T
1Þae1 þ Ze3 ða0 ; . . . ; ae3 ÞhðxÞa0 % % % % 2e3 T
ðx 1Þbe1 þ Ze3 ðb0 ; . . . ; be3 ÞhðxÞa0 mod 2; % % % %
that is, ðx2
e3
T
1Þðae1 þ be1 Þ % %
½Ze3 ða0 ; . . . ; ae3 Þ þ Ze3 ðb0 ; . . . ; be3 ÞhðxÞa0 mod 2: % % % % %
ð5Þ
e3
On the other hand, by x2 T 1 2e2 he2 ðxÞ mod f ðxÞ acting on a ¼ % a0 þ a1 2 þ þ ae1 2e1 ; we get % % % e3 ðx2 T 1Þðae2 þ ae1 2Þ2e2 2e2 he2 ðxÞða0 þ a1 2Þ; % % % % that is, ðx2
e3
T
1Þðae2 þ ae1 2Þ he2 ðxÞða0 þ a1 2Þ mod 22 : % % % %
ð6Þ
Let he2 ðxÞða0 þ a1 2Þ u þ v2 mod 22 , where u and v are the 0th and 1th % % of h % ðxÞð%a þ a 2Þ, respectively, % % level components then u hðxÞa0 mod 2. e2 0 1 % % % % By (6), e3
x2
T
ae2 þ ðx2 %
e3
T
1Þae1 2 ae2 þ u þ v2 mod 22 % % % %
and so e3
ðx2
T
1Þae1 v þ ae2 u mod 2: % % % %
ð7Þ
T
1Þbe1 w þ be2 u mod 2; % % % %
ð8Þ
Similarly, e3
ðx2
where he2 ðxÞðb0 þ b1 2Þ u þ w2 mod 22 and u he2 ðxÞb0 hðxÞa0 mod 2. % a% þ b Þ mod 2: By % (7) and %(8), % Since a0 ¼ b%0 , v þ% w hðxÞð 1 1 % % % % % % e3 ðx2 T 1Þðae1 þ be1 Þ ðae2 þ be2 ÞhðxÞa0 þ hðxÞða1 þ b1 Þ mod 2: ð9Þ % % % % % % % Comparing with (5), it follows that ðae2 þ be2 ÞhðxÞa0 % % %
hðxÞða1 þ b1 Þ½Ze3 ða0 ; . . . ; ae3 Þ þ Ze3 ðb0 ; . . . ; be3 ÞhðxÞa0 mod 2: % % % % % % %
ð10Þ
576
WENFENG AND XUANYONG e4
If e 5, then x2 T 1 2e3 he3 ðxÞ mod f ðxÞ acts on a ¼ a0 þ a1 2 þ % % % þ ae1 2e1 and b ¼ b0 þ b1 2 þ þ be1 2e1 . It follows that % % % % % e4 ð11Þ ðx2 T 1Þðae3 þ ae2 2Þ he3 ðxÞða0 þ a1 2Þ mod 22 ; % % % % e4 ð12Þ ðx2 T 1Þðbe3 þ be2 2Þ he3 ðxÞðb0 þ b1 2Þ mod 22 : % % % % Similar to (9), we get e4
ðx2
T
1Þðae2 þ be2 Þ ðae3 þ be3 ÞhðxÞa0 þ hðxÞða1 þ b1 Þ mod 2: % % % % % % %
Multiplying (13) by hðxÞa0 , since perðhðxÞa0 Þ divides T ; we obtain % % e4 ðx2 T 1Þ½ðae2 þ be2 ÞhðxÞa0 % % %
ðae3 þ be3 ÞhðxÞa0 þ hðxÞða1 þ b1 ÞhðxÞa0 mod 2: % % % % % %
ð13Þ
ð14Þ
By (10) and (3), we have ðae2 þ be2 ÞhðxÞa0 hðxÞða1 þ b1 Þ½ae3 Ze4 ða0 ; . . . ; ae4 Þ % % % % % % % % þ be3 Ze4 ðb0 ; . . . ; be4 Þ þ ce4 ða0 ; . . . ; ae4 Þ % % % % % þ ce4 ðb0 ; . . . ; be4 ÞhðxÞa0 mod 2: % % % It follows that e4
ðx2
T
1Þ½ðae2 þ be2 ÞhðxÞa0 % % % 2e4 T
ðx 1Þ½ae3 Ze4 ða0 ; . . . ; ae4 Þ þ be3 Ze4 ðb0 ; . . . ; be4 ÞhðxÞa0 % % % % % % %
½Ze4 ða0 ; . . . ; ae4 Þ þ Ze4 ðb0 ; . . . ; be4 ÞhðxÞa0 mod 2: % % % % %
Comparing with (14), we get ðae3 þ be3 ÞhðxÞa0 hðxÞða1 þ b1 ÞhðxÞa0 þ ½Ze4 ða0 ; . . . ; ae4 Þ % % % % % % % % þ Ze4 ðb0 ; . . . ; be4 ÞhðxÞa0 mod 2: % % %
ð15Þ
Now, we show hðxÞða1 þ b1 ÞhðxÞa0 hðxÞða1 þ b1 Þ mod 2. If a1 ¼ b1 , then % % % % % % % hðxÞða1 þ b1 ÞhðxÞa0 hðxÞða1 þ b1 Þ 0 mod 2: % % % % % If a1 =b1 , then 0th level component of a b is 0 and 1th level component % a % primitive sequence generated a % b is% a1 þ b1 mod 2. Thus a1 þ b1 is %by f% ðxÞ %over % F , so is hðxÞð%a þ%b Þ. Let hðxÞa ¼ ðu ; u ; u ; . . .Þ and 2 1 1 0 0 1 2 % % %
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
577
hðxÞða1 þ b1 Þ ¼ ðs0 ; s1 ; s2 ; . . .Þ over F2 , then by (10), si ¼ 0 if ui ¼ 0. And since%hðxÞð%a1 þ b1 Þ and hðxÞa0 are primitive sequences generated by f ðxÞ over F2 , we get% hðxÞa% 0 hðxÞða1 %þ b1 Þ mod 2, that is a1 þ b1 a0 mod 2. So % % % % % % hðxÞða1 þ b1 ÞhðxÞa0 hðxÞða1 þ b1 Þ mod 2: % % % % % Then, (15) implies ðae3 þ be3 ÞhðxÞa0 hðxÞða1 þ b1 Þ þ ½Ze4 ða0 ; . . . ; ae4 Þ % % % % % % % þ Ze4 ðb0 ; . . . ; be4 ÞhðxÞa0 mod 2: % % % In general, we have ðaei þ bei ÞhðxÞa0 hðxÞða1 þ b1 Þ þ ½Zei1 ða0 ; . . . ; aei1 Þ % % % % % % % þ Zei1 ðb0 ; . . . ; bei1 ÞhðxÞa0 mod 2; % % %
ð16Þ
where i ¼ 2; 3; . . . ; e 2: Finally, by xT 1 2h1 ðxÞ mod f ðxÞ acting on a % and b, similar to (9), we have % ðxT 1Þða2 þ b2 Þ ða1 þ b1 Þh1 ðxÞa0 þ h1 ðxÞða1 þ b1 Þ mod 2; ð17Þ % % % % % % % which implies ðxT 1Þ½ða2 þ b2 ÞhðxÞa0 ða1 þ b1 Þh1 ðxÞa0 hðxÞa0 % % % % % % % þ h1 ðxÞða1 þ b1 ÞhðxÞa0 mod 2: % % %
ð18Þ
On the other hand, by (16) in the case of i ¼ e 2, ða2 þ b2 ÞhðxÞa0 hðxÞða1 þ b1 Þ þ ½Z1 ða0 ; a1 Þ þ Z1 ðb0 ; b1 ÞhðxÞa0 mod 2 % % % % % % % % % % and Z1 ðx0 ; x1 Þ ¼ x1 Z0 ðx0 Þ þ c0 ðx0 Þ; we have ðxT 1Þ½ða2 þ b2 ÞhðxÞa0 % % % T
ðx 1Þ½ðZ1 ða0 ; a1 Þ þ Z1 ðb0 ; b1 ÞÞhðxÞa0 þ hðxÞða1 þ b1 Þ % % % % % % %
ðxT 1Þ½a1 Z0 ða0 Þ þ b1 Z0 ðb0 ÞhðxÞa0 % % % % %
Z1 ða0 ÞhðxÞa0 ðxT 1Þða1 þ b1 Þ % % % %
Z1 ða0 ÞhðxÞa0 ðh1 ðxÞa0 þ h1 ðxÞb0 Þ 0 mod 2: % % % % So (18) implies ða1 þ b1 Þh1 ðxÞa0 hðxÞa0 h1 ðxÞða1 þ b1 ÞhðxÞa0 mod 2: % % % % % % %
ð19Þ
578
WENFENG AND XUANYONG
If a1 =b1 ; then a1 þ b1 a0 mod 2 and the above equation implies % % % % % ð20Þ a0 h1 ðxÞa0 hðxÞa0 h1 ðxÞa0 hðxÞa0 mod 2: % % % % % It is clear that a0 ; h1 ðxÞa0 and hðxÞa0 are linear independent over F2 ; since % 2; h ðxÞc0 % mod 2 and hðxÞc0; 1 mod 2. So, by hðxÞ h1 ðxÞ þ h%1 ðxÞ2 mod 1 Lemma 4, (20) is not true. Thus a1 ¼ b1 and by (19), % by (16) again, we ða2 þ b2 ÞhðxÞa0 0 mod 2, which implies a2 ¼ b%2 : And % % % % % have ðaei þ bei ÞhðxÞa0 0 mod 2, which implies aei ¼ bei ; i ¼ e 3; % % ¼b % % e 4; . .% . ; 2: Finally ae1 e1 since ak ¼ bk ; k ¼ 0; 1; . . . ; e 2; and jða0 ; % % % % % . . . ; ae1 Þ jðb0 ; . . . ; be1 Þ mod 2: Therefore, a ¼ b: % % % % % Secondly, we consider the case e ¼ 3. We have jðx0 ; x1 ; x2 Þ ¼ x2 þ Zðx0 ; x1 Þ and Zðx0 ; x1 Þ ¼ x1 Z0 ðx0 Þ þ c0 ðx0 Þ. Since jða0 ; a1 ; a2 Þ jðb0 ; b1 ; b2 Þ and % % % % % % a0 ¼ b0 , it follows that % % ð21Þ a2 þ b2 ða1 þ b1 ÞZ0 ða0 Þ mod 2: % % % % % By Lemmas 3 and 2, ðxT 1Þða2 þ b2 Þ ðxT 1Þðða1 þ b1 ÞZ0 ða0 ÞÞ % % % % %
Z0 ða0 ÞðxT 1Þða1 þ b1 Þ % % %
Z0 ða0 Þðh1 ðxÞa0 þ h1 ðxÞb0 Þ 0 mod 2: % % % By (17), we have h1 ðxÞða1 þ b1 Þ ða1 þ b1 Þh1 ðxÞa0 mod 2: % % % % % If a1 þ b1 c0 mod 2, then a1 þ b1 mod 2 is a primitive sequence over % 2, % h ðxÞða þ b Þ and h ðxÞa are also F2 ; % and% since h1 ðxÞc0 mod 1 1 1 1 0 % %is in contradiction % primitive sequences over F2 . This condition with the above equation. So a1 þ b1 0 mod 2, and by (21), we have a2 ¼ b2 and % % % % a ¼ b. % Finally, % considering the case e ¼ 2, we have jðx0 ; x1 Þ ¼ x1 þ Zðx0 Þ. So j ða0 ; a1 Þ jðb0 ; b1 Þ mod 2 and a0 ¼ b0 imply a1 ¼ b1 . Hence a ¼ b: & % % % % % % % % % % 3.
COMPRESSION MAPPINGS OVER GALOIS RINGS
Let p be a prime, Zp the p-adic integer ring and Qp the p-adic number field. Let K be an unramified extension of Qp with degree r, R the integer ring of K, then GRðpe ; rÞ ¼ R=p e R is called a Galois ring, where e is a positive integer.
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
579
Remark 2. (1) Let gðxÞ be a monic polynomial over Z=ðp e Þ with degree r. If gðxÞ mod p is irreducible over Fp , then A½x=ðgðxÞÞ ffi GRðpe ; rÞ, where A ¼ Z=ðpe Þ. (2) GRðp e ; 1Þ ¼ Z=ðpe Þ. (3) GRðp e ; rÞ is a local ring with the maximal ideal ðpÞ ¼ pGRðpe ; rÞ ¼ fpa j a 2 GRðp e ; rÞg and GRðp; rÞ ¼ GRðpe ; rÞ=ðpÞ ¼ Fpr is a finite field of p r elements. r (4) Let O ¼ fa 2 GRðpe ; rÞjap ¼ ag, then O consists of pr elements, which are distinct modulo p. So the mapping O ! Fpr , a/a mod p is bijective. Furthermore, each element a in GRðpe ; rÞ may be written uniquely as a ¼ a0 þ a1 p þ þ ae1 pe1 ; where ai 2 O. We call O the p-adic coordinate set of GRðpe ; rÞ (see [5]). (5) Let a be a sequence over GRðpe ; rÞ, then a may be written uniquely as % % a ¼ a0 þ a1 p þ þ ae1 pe1 ; % % % % where ai ¼ ðai0 ; ai1 ; . . .Þ is a sequence over O, i ¼ 0; 1; . . . ; e 1: The sequence% ai is called ith level component of a and ae1 the highest level % % component% of a. % Now we set p ¼ 2 and let f ðxÞ be a monic polynomial over GRð2e ; rÞ. If f ð0Þc0 mod 2, then there exists a positive integer P such that f ðxÞ divides xP 1 and the least such P is called the period of f ðxÞ over GRð2e ; rÞ; denoted by perðf ðxÞÞ. For a monic polynomial f ðxÞ over GRð2e ; rÞ with degree n, the period of f ðxÞ is upper bounded by 2e1 ð2rn 1Þ and f ðxÞ is called a primitive polynomial if perðf ðxÞÞ ¼ 2e1 ð2rn 1Þ. Let f ðxÞ be a primitive polynomial of degree n over GRð2e ; rÞ; then f ðxÞ mod 2 is a primitive polynomial over F2r : Let a be a sequence over GRð2e ; rÞ; generated by a primitive polynomial f ðxÞ of %degree n with ac0 mod 2, then perðaÞ ¼ perðf ðxÞÞ ¼ 2e1 T and perðai Þ ¼ 2i T where T ¼ 2% rn 1, i ¼ 0; 1; . . . ; e % 1. Especially perðae1 Þ ¼ perðaÞ% ¼ 2e1 T . % % Lemma 6. Let f ðxÞ be a primitive polynomial of degree n over GRð2e ; rÞ, T ¼ 2rn 1; then there exists hi ðxÞ over GRð2e ; rÞ of degree less than n, i ¼ 1; 2; . . . ; e 1; such that x2
i1
T
1 2i hi ðxÞ mod f ðxÞ:
ð22Þ
Furthermore, all hi ðxÞc0 mod 2, h2 ðxÞ he1 ðxÞ mod 2 and h2 ðxÞ
h1 ðxÞ þ h1 ðxÞ2 modð2; f ðxÞÞ.
580
WENFENG AND XUANYONG
Definition 1. Let f ðxÞ be a primitive polynomial over GRð2e ; rÞ. If e 3 and degðh2 ðxÞ mod 2Þ 1 or e ¼ 2 and degðh1 ðxÞ mod 2Þ 1, then f ðxÞ is called a strongly primitive polynomial over GRð2e ; rÞ. Lemma 7. Let f ðxÞ be a primitive polynomial over GRð2e ; rÞ, Zðx0 ; x1 ; . . . ; xe2 Þ a function of e 1 variables over F2r and jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ: For a; b 2 Gðf ðxÞ; GRð2e ; rÞÞ; if jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; be1 Þ mod 2; then % % % % a0 ¼%b0%. % % The proof is similar to the one of Lemma 5. Lemma 8. Let O be the p-adic coordinate set of GRð2e ; rÞ, d 2 GRð2e ; rÞ, a; b 2 O; then 1. 2. 3. 4.
r1
ðabÞ2
r
d2 d mod 2. s For s e 1, d2 2 O. ab 2 O. rs Let s be a positive integer such that rs e 1; then ða þ bÞ2 2 O, 2 O and rs
r1
a þ b ða þ bÞ2 þ 2ðabÞ2
mod 22 : r
Proof. (1) Let d ¼ l þ 2x; where l 2 O, x 2 GRð2e ; rÞ. Since l2 ¼ l, d d mod 2. (2) If d 0 mod 2; then d ¼ 2x; where x 2 GRð2e ; rÞ. Since 2s 2e1 e; s we have d2 ¼ 0 2 O: If dc0 mod 2; then d ¼ l þ 2x; where l 2 O, s s s s s x 2 GRð2e ; rÞ. So d2 ¼ ðl þ 2xÞ2 ¼ l2 þ 21þs l2 1 x þ ¼ l2 2 O: r 2r 2r (3) ðabÞ ¼ a2 b ¼ ab; so ab 2 O. rs r1 (4) By (2) and (3), ða þ bÞ2 2 O; ðabÞ2 2 O: Since 2r
k
k
k
k1
k1
ða þ bÞ2 a2 þ b2 þ 2a2 b2
mod 22 ;
we have rs
rs
rs
ða þ bÞ2 a2 þ b2 þ 2a2
a þ b þ 2ða2
rs1
rðs1Þ
rs
r1
So a þ b ða þ bÞ2 þ 2ðabÞ2
mod 22 :
&
r1
rs1
Þ2 ðb2
r1
a þ b þ 2ðabÞ2
b2
rðs1Þ
mod 22 :
Þ2
r1
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
581
Lemma 9. Let f ðxÞ be a primitive polynomial of degree n over the finite field F2r , u ¼ ðu0 ; u1 ; . . .Þ, v ¼ ðv0 ; v1 ; . . .Þ 2 Gðf ðxÞ; F2r Þ and v=0. If vi ¼ 0 % 0, then there exists % % implies ui ¼ c 2 F2r such that u ¼ c v. % % Proof. If u ¼ 0, then we set c ¼ 0 and get u ¼ cv. Now assume u=0, % condition. Since f ðxÞ is a primitive % polynomial % % n then v=0 by the of degree % and u and v are generated by f ðxÞ, there exists a nonnegative integer k such that % ðuk ; %ukþ1 ; . . . ; ukþn1 Þ ¼ ð0; 0; . . . ; 0; aÞ and ðvk ; vkþ1 ; . . . ; vkþn1 Þ ¼ ð0; 0; . . . ; 0; bÞ, where a=0 and b=0: Then it is clear that u ¼ cv, where % % c ¼ ab1 : & Lemma 10. Let f ðxÞ be a primitive polynomial of degree n over GRð2e ; rÞ, T ¼ 2rn 1, a ¼ a0 þ a1 2 þ þ ae1 2e1 2 Gðf ðxÞ; GRð2e ; rÞÞ and a0 =0; % % Þ ¼ 2e1 T ¼ perðaÞ ¼ perðf ðxÞÞ. % then perðai Þ ¼%2i T ,% especially perðae1 % % % The proof is similar to one of Lemma 1 cited from [1]. Lemma 11. Let f ðxÞ be a strongly primitive polynomial over GRð2e ; rÞ, e 3, a; b 2 Gðf ðxÞ; GRð2e ; rÞÞ and a0 ¼ b0 =0. If there exists c 2 F2r such % b% ca mod 2 and % % that a1 þ 1 0 % % % a0 h1 ðxÞa0 ðh2 ðxÞa0 Þ2 cðh1 ðxÞða0 ÞÞ2 ðh2 ðxÞa0 Þ2 mod 2; ð23Þ % % % % % where h1 ðxÞ and h2 ðxÞ is defined by (22), then a1 ¼ b1 ; i.e. c ¼ 0: % % Proof. Assume a1 =b1 ; that is, c=0. If a0 , h1 ðxÞa0 and h% 2 ðxÞ%a0 are linear independent over F2r , then, by Lemma 4, % % % a0 h1 ðxÞa0 ðh2 ðxÞa0 Þ2 ccðh1 ðxÞða0 ÞÞ2 ðh2 ðxÞa0 Þ2 mod 2; % % % % % that is a contradiction. Now suppose that a0 , h1 ðxÞa0 and h2 ðxÞa0 are linear % % % over F r , dependent over F2r : Since a0 is an m-sequence generated by f ðxÞ 2 % deg ðh1 ðxÞ mod 2Þ 1, deg ðh2 ðxÞ mod 2Þ 1 and h1 ðxÞch2 ðxÞ mod 2, we conclude that any two of a0 , h1 ðxÞa0 and h2 ðxÞa0 are linear independent % a ¼ c% h ðxÞa þ c %h ðxÞa over F r , where c over F2r . Thus, we can assume 0 1 1 0 2 2 0 2 1 % in F r . % We write % a ¼ ða ; a ; a ; . . .Þ; and c2 are nonzero elements 2 0 0 1 2 c1 h1 ðxÞa0 ¼ ðb0 ; b1 ; b2 ; . . .Þ; c2 h2 ðxÞa0 ¼ ðg0 ; g1 ; g2 ; . . .Þ %over F2r . Since c1 h1 ðxÞa% 0 and c2 h2 ðxÞa0 are linear% independent over F2r ; there exists % nonnegative integer % t such that bt ¼ gt =0: Then yt =0 in 2 cðh1 ðxÞða0 ÞÞ ðh2 ðxÞa0 Þ2 ¼ ðy0 ; y1 ; y2 ; . . .Þ. But at ¼ bt þ gt ¼ 2bt ¼ 0, by (23), % we get a% contradiction. & Lemma 12. Let gðxÞ ¼ cn xn þ cn1 xn1 þ þ c0 2 F2r ½x, f ðxÞ ¼ c2n xn þ þ þ c20 and u a sequence over F2r . Then ðgðxÞuÞ2 ¼ f ðxÞu2 : % Especially, ððxk 1ÞuÞ2 ¼ %ðxk 1Þðu2 Þ for any positive integer k.% % % The proof is easy.
c2n1 xn1
582
WENFENG AND XUANYONG
Theorem 3. Let f ðxÞ be a strongly primitive polynomial of degree n over GRð2e ; rÞ, jðx0 ; x1 ; . . . ; xe1 Þ ¼ xe1 þ Zðx0 ; x1 ; . . . ; xe2 Þ; where Zðx0 ; x1 ; . . . ; xe2 Þ is a function of e 1 variables over F2r . Set Ze2 ðx0 ; x1 ; . . . ; xe2 Þ ¼ Zðx0 ; x1 ; . . . ; xe2 Þ: If Ze2 ðx0 ; x1 ; . . . ; xe2 Þ satisfies Ze2 ðx0 ; x1 ; . . . ; xe2 Þ ¼ xe2 Ze3 ðx0 ; x1 ; . . . ; xe3 Þ þ ce3 ðx0 ; x1 ; . . . ; xe3 Þ and Zi ðx0 ; x1 ; . . . ; xi Þ ¼ xi Zi1 ðx0 ; x1 ; . . . ; xi1 Þ þ ci1 ðx0 ; x1 ; . . . ; xi1 Þ; where i ¼ 1; 2; . . . ; e 2, Zi1 ðx0 ; x1 ; . . . ; xi1 Þ and ci1 ðx0 ; x1 ; . . . ; xi1 Þ are functions of i variables over F2r , then the compression mapping ( Gðf ðxÞ; GRð2e ; rÞÞ ! F21 r ; F: a ¼ a0 þ a1 2 þ þ ae1 2e1 /jða0 ; a1 ; . . . ; ae1 Þ mod 2 % % % % % % % is injective, that is, for a; b 2 Gðf ðxÞ; GRð2e ; rÞÞ, a ¼ b if and only if % Þ mod 2. % % jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; b%e1 % % % % Proof. Let a; b 2 Gðf ðxÞ; GRð2e ; rÞÞ satisfying % % jða0 ; . . . ; ae1 Þ jðb0 ; . . . ; be1 Þ mod 2: % % % % By Lemma 7, a0 ¼ b0 . It is not harmful to assume a0 =0. % % ðxÞ mod 2; (1) Assume e% 4. Since h2 ðxÞ he1 hðxÞ ¼ hi ðxÞ mod 2 over F2r , 2 i e 1. Since
we
ae1 þ ae2 Ze3 ða0 ; . . . ; ae3 Þ þ ce3 ða0 ; . . . ; ae3 Þ % % % % % %
be1 þ be2 Ze3 ðb0 ; . . . ; be3 Þ þ ce3 ðb0 ; . . . ; be3 Þ mod 2 % % % % % %
set
ð24Þ
and by Lemma 10 the periods of ce3 ða0 ; . . . ; ae3 Þ and ce3 ðb0 ; . . . ; be3 Þ e3 % % divide 2e3 T , where T ¼ 2rn 1, by x2 T% 1 acting on (24), we% can get ðx2
e3
T
1Þðae1 þ ae2 Ze3 ða0 ; . . . ; ae3 ÞÞ % % % %
ðx2 T 1Þðbe1 þ be2 Ze3 ðb0 ; . . . ; be3 ÞÞ mod 2: % % % % e3
ð25Þ
Furthermore, by Lemma 3, we have e3
ðx2
e3
1Þae1 þ Ze3 ða0 ; . . . ; ae3 Þðx2 T 1Þae2 % % % % 2e3 T 2e3 T
ðx 1Þbe1 þ Ze3 ðb0 ; . . . ; be3 Þðx 1Þbe2 mod 2: % % % % T
ð26Þ
583
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
Since a0 ¼ b0 and by Lemma 6 x2 % that % ðx2
e3
T
e3
T
1 2e2 he2 ðxÞ mod f ðxÞ, it follows
1Þae2 hðxÞa0 hðxÞb0 ðx2 % % %
e3
T
1Þbe2 mod 2: %
So (26) implies that ðx2
e3
T
1Þðae1 þ be1 Þ % %
½Ze3 ða0 ; . . . ; ae3 Þ þ Ze3 ðb0 ; . . . ; be3 ÞhðxÞa0 mod 2: % % % % %
ð27Þ
e3
By x2 T 1 2e2 he2 ðxÞ mod f ðxÞ acting on a ¼ a0 þ a1 2 þ þ % % % ae1 2e1 , % e3 ðx2 T 1Þðae2 þ ae1 2Þ he2 ðxÞða0 þ a1 2Þ mod 22 : ð28Þ % % % % Let he2 ðxÞða0 þ a1 2Þ u þ v2 mod 22 ; % % % %
ð29Þ
where u and v are 0th and 1th level components of he2 ðxÞða0 þ a1 2Þ, % % % respectively. It %is clear that u he2 ðxÞa0 mod 2. So, by (28) % % e3 e3 x2 T ae2 þ ðx2 T 1Þae1 2 ae2 þ u þ v2 mod 22 : ð30Þ % % % % rs
r1
By Lemma 8, ae2 þ u ðae2 þ uÞ2 þ 2ðae2 uÞ2 mod 22 , where s is a % rs % e 1. % Since ða% %þ uÞ2rs is a sequence over O; % positive integer such that e2 % % by (30), it follows that ðx2
e3
r1
T
1Þae1 v þ ðuae2 Þ2 % % %%
T
1Þbe1 w þ ðube2 Þ2 % % %%
mod 2:
ð31Þ
mod 2;
ð32Þ
Similarly, we have ðx2
e3
r1
where w is determined by % he2 ðxÞðb0 þ b1 2Þ u þ w2 mod 22 : % % % % Since a0 ¼ b0 ; by (29) and (33), it follows % 2, and by (31) and (32), we have hðxÞða1 þ b%1 Þ mod % % e3
ðx2
T
r1
r1
1Þðae1 þ be1 Þ u2 ðae2 þ be2 Þ2 % % % % %
ð33Þ that
vþw¼ % %
þ hðxÞða1 þ b1 Þ mod 2: % %
ð34Þ
584
WENFENG AND XUANYONG
By (27) and u hðxÞa0 mod 2, % % r1
u2 ðae2 þ be2 Þ2 % % %
r1
½Ze3 ða0 ; . . . ; ae3 Þ % % þ Ze3 ðb0 ; . . . ; be3 Þu þ hðxÞða1 þ b1 Þ mod 2: % % % % %
ð35Þ
Then r1
r1
uðae2 þ be2 Þ ½u2 ðae2 þ be2 Þ2 2 % % % % % %
½Ze3 ða0 ; . . . ; ae3 Þ þ Ze3 ðb0 ; . . . ; be3 Þ2 u2 % % % % % þ ½hðxÞða1 þ b1 Þ2 mod 2: % % If e 5, x2 Then
e4
T
ð36Þ
1 2e3 he3 ðxÞ mod f ðxÞ acts on a and b continuously. % %
e4
T
1Þðae3 þ ae2 2Þ he3 ðxÞða0 þ a1 2Þ mod 22 ; % % % %
e4
T
1Þðbe3 þ be2 2Þ he3 ðxÞðb0 þ b1 2Þ mod 22 : % % % %
ðx2 ðx2
Similar to (34), we have ðx2
e4
T
r1
r1
1Þðae2 þ be2 Þ u2 ðae3 þ be3 Þ2 % % % % %
þ hðxÞða1 þ b1 Þ mod 2 % % ð37Þ
and so ½ðx2
e4
T
r1
r1
1Þðae2 þ be2 Þu u½u2 ðae3 þ be3 Þ2 % % % % % % %
þ hðxÞða1 þ b1 Þ mod 2: % %
Since perðuÞ ¼ T ; we have % ðx2
e4
T
1Þ½ðae2 þ be2 Þu % % % r1
uu2 ðae3 þ be3 Þ2 % % %%
r1
þ uhðxÞða1 þ b1 Þ mod 2: % % %
ð38Þ
585
COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
And by (36), it follows that ðae2 þ be2 Þu % % %
u2 ½ae3 Ze4 ða0 ; . . . ; ae4 Þ þ be3 Ze4 ðb0 ; . . . ; be4 Þ % % % % % % % þ ce4 ða0 ; . . . ; ae4 Þ þ ce4 ðb0 ; . . . ; be4 Þ2 þ ½hðxÞða1 þ b1 Þ2 mod 2: % % % % % % The periods of ðce4 ða0 ; . . . ; ae4 Þ þ ce4 ðb0 ; . . . ; be4 ÞÞ2 u2 and ðhðxÞða1 þ % % % % b1 ÞÞ2 divide 2e4 T , so it%follows% that % e4 ðx2 T 1Þ½ðae2 þ be2 Þu % % %
ðx2
e4
T
1Þ½ae3 Ze4 ða0 ; . . . ; ae4 Þ þ be3 Ze4 ðb0 ; . . . ; be4 Þ2 u2 mod 2: % % % % % % %
Comparing with (38) and by Lemma 12 we get e4
½Ze4 ða0 ; . . . ; ae4 Þðx2 % % r1
T
u u2 ðae3 þ be3 Þ2 % % % % Since ðx2
e4
T
e4
1Þae3 þ Ze4 ðb0 ; . . . ; be4 Þðx2 % % % r1
T
1Þbe3 2 u2 % %
þ u hðxÞða1 þ b1 Þ mod 2: % % %
1Þae3 he3 ðxÞa0 u mod 2, then % % % ½Ze4 ða0 ; . . . ; ae4 Þ þ Ze4 ðb0 ; . . . ; be4 Þ2 u4 % % % % % r1
u u2 ðae3 þ be3 Þ2 % % % %
r1
þ u hðxÞða1 þ b1 Þ mod 2: % % %
So ½Ze4 ða0 ; . . . ; ae4 Þ þ Ze4 ðb0 ; . . . ; be4 Þ4 u8 % % % % % r1
r1
½u u2 ðae3 þ be3 Þ2 % % % %
þ u hðxÞða1 þ b1 Þ2 % % %
u2 uðae3 þ be3 Þ þ u2 ½hðxÞða1 þ b1 Þ2 mod 2; % % % % % % % that is, uðae3 þ be3 Þu2 ½Ze4 ða0 ; . . . ; ae4 Þ þ Ze4 ðb0 ; . . . ; be4 Þ4 u8 % % % % % % % % % 2 2 þ u ½hðxÞða1 þ b1 Þ mod 2: % % %
ð39Þ
Let u ðu0 ; u1 ; . . .Þ mod 2; hðxÞða1 þ b1 Þ ¼ ðw0 ; w1 ; . . .Þ mod 2; by (36), it % by% Lemma 9, hðxÞða þ b Þ cu mod 2 follows%that if ui ¼ 0; then wi 0: So 1 1 % % %
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for some c 2 F2r ; and (39) implies uðae3 þ be3 Þ u6 ½Ze4 ða0 ; . . . ; ae4 Þ þ Ze4 ðb0 ; . . . ; be4 Þ4 % % % % % % % % þ ½hðxÞða1 þ b1 Þ2 mod 2: % % From the above discussion, we deduce the following formula: uðaei þ bei Þ uki ðZei1 ða0 ; . . . ; aei1 Þ þ Zei1 ðb0 ; . . . ; bei1 ÞÞ2 % % % % % % % % þ ½hðxÞða1 þ b1 Þ2 mod 2; % %
i1
ð40Þ
where ki is a positive integer, i ¼ 2; 3; . . . ; e 2. Take i ¼ e 2; then uða2 þ b2 Þ uke2 ½Z1 ða0 ; a1 Þ þ Z1 ðb0 ; b1 Þ2 % % % % % % % %
e3
þ ½hðxÞða1 þ b1 Þ2 mod 2: % %
ð41Þ
Finally, xT 1 2h1 ðxÞ mod f ðxÞ acts on a and b. Similar to (34), we have % % r1
r1
ðxT 1Þða2 þ b2 Þ ðh1 ðxÞa0 Þ2 ða1 þ b1 Þ2 % % % % %
þ h1 ðxÞða1 þ b1 Þ mod 2; % %
ð42Þ
which deduces r1
r1
ðxT 1Þða2 þ b2 Þu ðh1 ðxÞa0 Þ2 ða1 þ b1 Þ2 u þ h1 ðxÞða1 þ b1 Þu mod 2:ð43Þ % % % % % % % % % % Since Z1 ðx0 ; x1 Þ ¼ x1 Z0 ðx0 Þ þ c0 ðx0 Þ and hðxÞða1 þ b1 Þ cu mod 2, (41) im% % % plies e3
ðxT 1Þða1 Z0 ða0 Þ þ b1 Z0 ðb0 ÞÞ2 uke2 % % % % % 2r1 2r1
ðh1 ðxÞa0 Þ ða1 þ b1 Þ u þ h1 ðxÞða1 þ b1 Þu mod 2: % % % % % % %
ð44Þ
Since ðxT 1Þa1 ðxT 1Þb1 h1 ðxÞa0 mod 2 and Z0 ða0 Þ Z0 ðb0 Þ mod 2; we % % % % % have e3
ðxT 1Þða1 Z0 ða0 Þ þ b1 Z0 ðb0 ÞÞ2 uke2 % % % % % e3
½ðxT 1Þa1 Z0 ða0 Þ þ ðxT 1Þb1 Z0 ðb0 Þ2 uke2 0 mod 2: % % % % %
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COMPRESSIVE MAPPINGS ON PRIMITIVE SEQUENCES
So (44) implies r1
r1
ðh1 ðxÞa0 Þ2 ða1 þ b1 Þ2 u þ h1 ðxÞða1 þ b1 Þu 0 mod 2 % % % % % % % and then r1
r1
ðh1 ðxÞða1 þ b1 ÞÞ2 u2 ½ðh1 ðxÞa0 Þ2 ða1 þ b1 Þ2 u2 % % % % % % %
h1 ðxÞa0 ða1 þ b1 Þu2 mod 2: % % % %
ð45Þ
Because hðxÞða1 þ b1 Þ cu mod 2 and h2 ðxÞa0 u mod 2, hðxÞða1 þ b1 Þ
% % % % that chðxÞa0 hðxÞðca%0 Þ; which implies a1 þ b1 ca0% mod% 2. And (45) implies % % % % % a0 h1 ðxÞa0 ðh2 ðxÞa0 Þ2 cðh1 ðxÞða0 ÞÞ2 ðh2 ðxÞa0 Þ2 mod 2: % % % % % By Lemma 11, we get a1 ¼ b1 . Thus uða2 þ b2 Þ 0 mod 2 by (41). Since % that % a þ b %is% a primitive % a0 ¼ b0 ; a1 ¼ b1 , it follows sequence over F2r or 2 2 %zero sequence. % % %And since u is a primitive % % sequence over F r ; we get a þ b
2 2 2 % by (40), it follows that a ¼ b ; j ¼ 3; .%. . ; e % 2: 0 mod 2; that is, a2 ¼ b2 . So j j % % % Þ mod 2, we have Finally, by the condition jða0 ; . . . ; ae1 Þ jðb0 ; . . %. ; be1 % % % % ae1 ¼ be1 ; and then a ¼ b. % (2) % Assume e%¼ 3;% then jðx0 ; x1 ; x2 Þ ¼ x2 þ Zðx0 ; x1 Þ and Zðx0 ; x1 Þ ¼ x1 Z0 ðx0 Þ þ c0 ðx0 Þ. Since jða0 ; a1 ; a2 Þ jðb0 ; b1 ; b2 Þ and a0 ¼ b0 , % % % % % % % % it follows that a2 þ b2 ða1 þ b1 ÞZ0 ða0 Þ mod 2: % % % % %
ð46Þ
By Lemmas 3 and 6, ðxT 1Þða2 þ b2 Þ ðxT 1Þðða1 þ b1 ÞZ0 ða0 ÞÞ % % % % %
Z0 ða0 ÞðxT 1Þða1 þ b1 Þ % % %
Z0 ða0 Þðh1 ðxÞa0 þ h1 ðxÞb0 Þ % % %
0 mod 2: r1
r1
By (42), we have h1 ðxÞða1 þ b1 Þ ða1 þ b1 Þ2 ðh1 ðxÞa0 Þ2 mod 2, and so % % % % % r1 r1 ðh1 ðxÞða1 þ b1 ÞÞ2 ½ða1 þ b1 Þ2 ðh1 ðxÞa0 Þ2 2 % % % % %
ða1 þ b1 Þh1 ðxÞa0 mod 2: % % % If a1 þ b1 c0 mod 2, that is, a1 þ b1 mod 2 and h1 ðxÞða1 þ b1 Þ are % % sequences over F %r . %Since degðh ðxÞ mod% 2Þ % 1; it primitive 2 1
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WENFENG AND XUANYONG
follows that and a1 þ b1 0 mod 2; Furthermore, by (46), a2 ¼ b2 % % % % and a ¼ b. % % (3) Assume e ¼ 2; then jðx0 ; x1 Þ ¼ x1 þ Zðx0 Þ. So jða0 ; a1 Þ jðb0 ; b1 Þ mod % % % % 2 and a0 ¼ b0 imply a1 ¼ b1 . Hence a ¼ b: &. % % % % % % Remark 3. Theorems 2 and 3 show that the binary sequence jða0 ; . . . ; % for ae1 Þ contains all information of the original sequence a. We guess that % any Z, Theorem 3 is also correct. REFERENCES 1. Z. D. Dai, Binary sequences derived from ML-sequences over rings I. Periods and minimal polynomials, J. Cryptol. 5, No. 4 (1992), 193–207. 2. Z. D. Dai, T. Beth, and D. Gollman, Lower bounds for the linear complexity of sequences over residue rings, in ‘‘Advances in Cryptology}EUROCRYPT’90’’ (I. B. Damgard Ed.), Lecture Notes in Computer Science, Vol. 473, pp. 189–195, Springer-Verlag, Berlin, 1991. 3. M. Q. Huang, ‘‘Analysis and Cryptologic Evaluation of Primitive Sequences over an Integer Residue Ring,’’ Doctoral dissertation of Graduate School of USTC, Academia Sinica, 1988. 4. M. Q. Huang and Z. D. Dai, Projective maps of linear recurring sequences with maximal p-adic periods, Fibonacci Quart. 30, No. 2 (1992), 139–143. 5. V. L. Kurakin, The first coordinate sequence of a linear recurrence of maximal period over a Galois ring, Discrete Math. Appl. 4, No. 2 (1994), 129–141. 6. A. S. Kuzmin and A. A. Nechaev, ‘‘A Construction of Noise Stable Codes using Linear Recurrences Over Galois Rings,’’ Russian Mathematical Survey, Vol. 47, pp. 189–190, 1992. 7. A. S. Kuzmin and A. A. Nechaev, ‘‘Linear Recurring Sequences over Galois Rings,’’ Russian Mathematical Survey, Vol. 48, pp. 171–172, 1993. 8. Qi Wenfeng, Yang Junhui, and Zhou Jinjun, ML-sequences over rings Z=ð2e Þ, in ‘‘Advances in Cryptology}ASIACRYPT’98,’’ (K. Ohta and D. Pei, Eds.), Lecture Notes in Computer Science, Vol. 1514, pp. 315–325, Springer-Verlag, Berlin, Heidelberg, 1998.